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Hypothesis Testing

About hypothesis testing.

Contents (Click to skip to the section):

What is a Hypothesis?

What is hypothesis testing.

• Hypothesis Testing Examples (One Sample Z Test).
• Hypothesis Test on a Mean (TI 83).

Bayesian Hypothesis Testing.

• More Hypothesis Testing Articles
• Hypothesis Tests in One Picture
• Critical Values

What is the Null Hypothesis?

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A hypothesis is an educated guess about something in the world around you. It should be testable, either by experiment or observation. For example:

• A new medicine you think might work.
• A way of teaching you think might be better.
• A possible location of new species.
• A fairer way to administer standardized tests.

It can really be anything at all as long as you can put it to the test.

What is a Hypothesis Statement?

If you are going to propose a hypothesis, it’s customary to write a statement. Your statement will look like this: “If I…(do this to an independent variable )….then (this will happen to the dependent variable ).” For example:

• If I (decrease the amount of water given to herbs) then (the herbs will increase in size).
• If I (give patients counseling in addition to medication) then (their overall depression scale will decrease).
• If I (give exams at noon instead of 7) then (student test scores will improve).
• If I (look in this certain location) then (I am more likely to find new species).

A good hypothesis statement should:

• Include an “if” and “then” statement (according to the University of California).
• Include both the independent and dependent variables.
• Be testable by experiment, survey or other scientifically sound technique.
• Be based on information in prior research (either yours or someone else’s).
• Have design criteria (for engineering or programming projects).

Hypothesis testing can be one of the most confusing aspects for students, mostly because before you can even perform a test, you have to know what your null hypothesis is. Often, those tricky word problems that you are faced with can be difficult to decipher. But it’s easier than you think; all you need to do is:

• Figure out your null hypothesis,
• State your null hypothesis,
• Choose what kind of test you need to perform,
• Either support or reject the null hypothesis .

If you trace back the history of science, the null hypothesis is always the accepted fact. Simple examples of null hypotheses that are generally accepted as being true are:

• DNA is shaped like a double helix.
• There are 8 planets in the solar system (excluding Pluto).
• Taking Vioxx can increase your risk of heart problems (a drug now taken off the market).

How do I State the Null Hypothesis?

You won’t be required to actually perform a real experiment or survey in elementary statistics (or even disprove a fact like “Pluto is a planet”!), so you’ll be given word problems from real-life situations. You’ll need to figure out what your hypothesis is from the problem. This can be a little trickier than just figuring out what the accepted fact is. With word problems, you are looking to find a fact that is nullifiable (i.e. something you can reject).

Hypothesis Testing Examples #1: Basic Example

A researcher thinks that if knee surgery patients go to physical therapy twice a week (instead of 3 times), their recovery period will be longer. Average recovery times for knee surgery patients is 8.2 weeks.

The hypothesis statement in this question is that the researcher believes the average recovery time is more than 8.2 weeks. It can be written in mathematical terms as: H 1 : μ > 8.2

Next, you’ll need to state the null hypothesis .  That’s what will happen if the researcher is wrong . In the above example, if the researcher is wrong then the recovery time is less than or equal to 8.2 weeks. In math, that’s: H 0 μ ≤ 8.2

Rejecting the null hypothesis

Ten or so years ago, we believed that there were 9 planets in the solar system. Pluto was demoted as a planet in 2006. The null hypothesis of “Pluto is a planet” was replaced by “Pluto is not a planet.” Of course, rejecting the null hypothesis isn’t always that easy— the hard part is usually figuring out what your null hypothesis is in the first place.

Hypothesis Testing Examples (One Sample Z Test)

The one sample z test isn’t used very often (because we rarely know the actual population standard deviation ). However, it’s a good idea to understand how it works as it’s one of the simplest tests you can perform in hypothesis testing. In English class you got to learn the basics (like grammar and spelling) before you could write a story; think of one sample z tests as the foundation for understanding more complex hypothesis testing. This page contains two hypothesis testing examples for one sample z-tests .

One Sample Hypothesis Testing Example: One Tailed Z Test

A principal at a certain school claims that the students in his school are above average intelligence. A random sample of thirty students IQ scores have a mean score of 112.5. Is there sufficient evidence to support the principal’s claim? The mean population IQ is 100 with a standard deviation of 15.

Step 1: State the Null hypothesis . The accepted fact is that the population mean is 100, so: H 0 : μ = 100.

Step 2: State the Alternate Hypothesis . The claim is that the students have above average IQ scores, so: H 1 : μ > 100. The fact that we are looking for scores “greater than” a certain point means that this is a one-tailed test.

Step 4: State the alpha level . If you aren’t given an alpha level , use 5% (0.05).

Step 5: Find the rejection region area (given by your alpha level above) from the z-table . An area of .05 is equal to a z-score of 1.645.

Step 6: If Step 6 is greater than Step 5, reject the null hypothesis. If it’s less than Step 5, you cannot reject the null hypothesis. In this case, it is more (4.56 > 1.645), so you can reject the null.

One Sample Hypothesis Testing Examples: #3

Blood glucose levels for obese patients have a mean of 100 with a standard deviation of 15. A researcher thinks that a diet high in raw cornstarch will have a positive or negative effect on blood glucose levels. A sample of 30 patients who have tried the raw cornstarch diet have a mean glucose level of 140. Test the hypothesis that the raw cornstarch had an effect.

• State the null hypothesis : H 0 :μ=100
• State the alternate hypothesis : H 1 :≠100
• State your alpha level. We’ll use 0.05 for this example. As this is a two-tailed test, split the alpha into two. 0.05/2=0.025
• Find the z-score associated with your alpha level . You’re looking for the area in one tail only . A z-score for 0.75(1-0.025=0.975) is 1.96. As this is a two-tailed test, you would also be considering the left tail (z = 1.96)
•   If Step 5 is less than -1.96 or greater than 1.96 (Step 3), reject the null hypothesis . In this case, it is greater, so you can reject the null.

*This process is made much easier if you use a TI-83 or Excel to calculate the z-score (the “critical value”). See:

• Critical z value TI 83
• Z Score in Excel

Hypothesis Testing Examples: Mean (Using TI 83)

You can use the TI 83 calculator for hypothesis testing, but the calculator won’t figure out the null and alternate hypotheses; that’s up to you to read the question and input it into the calculator.

Example problem : A sample of 200 people has a mean age of 21 with a population standard deviation (σ) of 5. Test the hypothesis that the population mean is 18.9 at α = 0.05.

Step 1: State the null hypothesis. In this case, the null hypothesis is that the population mean is 18.9, so we write: H 0 : μ = 18.9

Step 2: State the alternative hypothesis. We want to know if our sample, which has a mean of 21 instead of 18.9, really is different from the population, therefore our alternate hypothesis: H 1 : μ ≠ 18.9

Step 3: Press Stat then press the right arrow twice to select TESTS.

Step 4: Press 1 to select 1:Z-Test… . Press ENTER.

Step 5: Use the right arrow to select Stats .

Step 6: Enter the data from the problem: μ 0 : 18.9 σ: 5 x : 21 n: 200 μ: ≠μ 0

Step 7: Arrow down to Calculate and press ENTER. The calculator shows the p-value: p = 2.87 × 10 -9

This is smaller than our alpha value of .05. That means we should reject the null hypothesis .

Bayesian Hypothesis Testing: What is it?

Bayesian hypothesis testing helps to answer the question: Can the results from a test or survey be repeated? Why do we care if a test can be repeated? Let’s say twenty people in the same village came down with leukemia. A group of researchers find that cell-phone towers are to blame. However, a second study found that cell-phone towers had nothing to do with the cancer cluster in the village. In fact, they found that the cancers were completely random. If that sounds impossible, it actually can happen! Clusters of cancer can happen simply by chance . There could be many reasons why the first study was faulty. One of the main reasons could be that they just didn’t take into account that sometimes things happen randomly and we just don’t know why.

It’s good science to let people know if your study results are solid, or if they could have happened by chance. The usual way of doing this is to test your results with a p-value . A p value is a number that you get by running a hypothesis test on your data. A P value of 0.05 (5%) or less is usually enough to claim that your results are repeatable. However, there’s another way to test the validity of your results: Bayesian Hypothesis testing. This type of testing gives you another way to test the strength of your results.

Traditional testing (the type you probably came across in elementary stats or AP stats) is called Non-Bayesian. It is how often an outcome happens over repeated runs of the experiment. It’s an objective view of whether an experiment is repeatable. Bayesian hypothesis testing is a subjective view of the same thing. It takes into account how much faith you have in your results. In other words, would you wager money on the outcome of your experiment?

Differences Between Traditional and Bayesian Hypothesis Testing.

Traditional testing (Non Bayesian) requires you to repeat sampling over and over, while Bayesian testing does not. The main different between the two is in the first step of testing: stating a probability model. In Bayesian testing you add prior knowledge to this step. It also requires use of a posterior probability , which is the conditional probability given to a random event after all the evidence is considered.

Arguments for Bayesian Testing.

Many researchers think that it is a better alternative to traditional testing, because it:

• Includes prior knowledge about the data.
• Takes into account personal beliefs about the results.

Arguments against.

• Including prior data or knowledge isn’t justifiable.
• It is difficult to calculate compared to non-Bayesian testing.

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Hypothesis Testing Articles

• What is Ad Hoc Testing?
• Composite Hypothesis Test
• What is a Rejection Region?
• What is a Two Tailed Test?
• How to Decide if a Hypothesis Test is a One Tailed Test or a Two Tailed Test.
• How to Decide if a Hypothesis is a Left Tailed Test or a Right-Tailed Test.
• How to State the Null Hypothesis in Statistics.
• How to Find a Critical Value .
• How to Support or Reject a Null Hypothesis.

Specific Tests:

• Brunner Munzel Test (Generalized Wilcoxon Test).
• Chi Square Test for Normality.
• Cochran-Mantel-Haenszel Test.
• Granger Causality Test .
• Hotelling’s T-Squared.
• KPSS Test .
• What is a Likelihood-Ratio Test?
• Log rank test .
• MANCOVA Assumptions.
• MANCOVA Sample Size.
• Marascuilo Procedure
• Rao’s Spacing Test
• Rayleigh test of uniformity.
• Sequential Probability Ratio Test.
• How to Run a Sign Test.
• T Test: one sample.
• T-Test: Two sample .
• Welch’s ANOVA .
• Welch’s Test for Unequal Variances .
• Z-Test: one sample .
• Z Test: Two Proportion.
• Wald Test .

Related Articles:

• What is an Acceptance Region?
• How to Calculate Chebyshev’s Theorem.
• Contrast Analysis
• Decision Rule.
• Degrees of Freedom .
• Directional Test
• False Discovery Rate
• How to calculate the Least Significant Difference.
• Levels in Statistics.
• How to Calculate Margin of Error.
• Mean Difference (Difference in Means)
• The Multiple Testing Problem .
• What is the Neyman-Pearson Lemma?
• What is an Omnibus Test?
• One Sample Median Test .
• How to Find a Sample Size (General Instructions).
• Sig 2(Tailed) meaning in results
• What is a Standardized Test Statistic?
• How to Find Standard Error
• Standardized values: Example.
• How to Calculate a T-Score.
• T-Score Vs. a Z.Score.
• Testing a Single Mean.
• Unequal Sample Sizes.
• Uniformly Most Powerful Tests.
• How to Calculate a Z-Score.

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Statistics By Jim

Making statistics intuitive

Hypothesis Testing: Uses, Steps & Example

By Jim Frost 4 Comments

What is Hypothesis Testing?

Hypothesis testing in statistics uses sample data to infer the properties of a whole population . These tests determine whether a random sample provides sufficient evidence to conclude an effect or relationship exists in the population. Researchers use them to help separate genuine population-level effects from false effects that random chance can create in samples. These methods are also known as significance testing.

For example, researchers are testing a new medication to see if it lowers blood pressure. They compare a group taking the drug to a control group taking a placebo. If their hypothesis test results are statistically significant, the medication’s effect of lowering blood pressure likely exists in the broader population, not just the sample studied.

Using Hypothesis Tests

A hypothesis test evaluates two mutually exclusive statements about a population to determine which statement the sample data best supports. These two statements are called the null hypothesis and the alternative hypothesis . The following are typical examples:

• Null Hypothesis : The effect does not exist in the population.
• Alternative Hypothesis : The effect does exist in the population.

Hypothesis testing accounts for the inherent uncertainty of using a sample to draw conclusions about a population, which reduces the chances of false discoveries. These procedures determine whether the sample data are sufficiently inconsistent with the null hypothesis that you can reject it. If you can reject the null, your data favor the alternative statement that an effect exists in the population.

Statistical significance in hypothesis testing indicates that an effect you see in sample data also likely exists in the population after accounting for random sampling error , variability, and sample size. Your results are statistically significant when the p-value is less than your significance level or, equivalently, when your confidence interval excludes the null hypothesis value.

Conversely, non-significant results indicate that despite an apparent sample effect, you can’t be sure it exists in the population. It could be chance variation in the sample and not a genuine effect.

Learn more about Failing to Reject the Null .

5 Steps of Significance Testing

Hypothesis testing involves five key steps, each critical to validating a research hypothesis using statistical methods:

• Formulate the Hypotheses : Write your research hypotheses as a null hypothesis (H 0 ) and an alternative hypothesis (H A ).
• Data Collection : Gather data specifically aimed at testing the hypothesis.
• Conduct A Test : Use a suitable statistical test to analyze your data.
• Make a Decision : Based on the statistical test results, decide whether to reject the null hypothesis or fail to reject it.
• Report the Results : Summarize and present the outcomes in your report’s results and discussion sections.

While the specifics of these steps can vary depending on the research context and the data type, the fundamental process of hypothesis testing remains consistent across different studies.

Let’s work through these steps in an example!

Hypothesis Testing Example

Researchers want to determine if a new educational program improves student performance on standardized tests. They randomly assign 30 students to a control group , which follows the standard curriculum, and another 30 students to a treatment group, which participates in the new educational program. After a semester, they compare the test scores of both groups.

Download the CSV data file to perform the hypothesis testing yourself: Hypothesis_Testing .

The researchers write their hypotheses. These statements apply to the population, so they use the mu (μ) symbol for the population mean parameter .

• Null Hypothesis (H 0 ) : The population means of the test scores for the two groups are equal (μ 1 = μ 2 ).
• Alternative Hypothesis (H A ) : The population means of the test scores for the two groups are unequal (μ 1 ≠ μ 2 ).

Choosing the correct hypothesis test depends on attributes such as data type and number of groups. Because they’re using continuous data and comparing two means, the researchers use a 2-sample t-test .

Here are the results.

The treatment group’s mean is 58.70, compared to the control group’s mean of 48.12. The mean difference is 10.67 points. Use the test’s p-value and significance level to determine whether this difference is likely a product of random fluctuation in the sample or a genuine population effect.

Because the p-value (0.000) is less than the standard significance level of 0.05, the results are statistically significant, and we can reject the null hypothesis. The sample data provides sufficient evidence to conclude that the new program’s effect exists in the population.

Limitations

Hypothesis testing improves your effectiveness in making data-driven decisions. However, it is not 100% accurate because random samples occasionally produce fluky results. Hypothesis tests have two types of errors, both relating to drawing incorrect conclusions.

• Type I error: The test rejects a true null hypothesis—a false positive.
• Type II error: The test fails to reject a false null hypothesis—a false negative.

Learn more about Type I and Type II Errors .

Our exploration of hypothesis testing using a practical example of an educational program reveals its powerful ability to guide decisions based on statistical evidence. Whether you’re a student, researcher, or professional, understanding and applying these procedures can open new doors to discovering insights and making informed decisions. Let this tool empower your analytical endeavors as you navigate through the vast seas of data.

Learn more about the Hypothesis Tests for Various Data Types .

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Reader Interactions

June 10, 2024 at 10:51 am

Thank you, Jim, for another helpful article; timely too since I have started reading your new book on hypothesis testing and, now that we are at the end of the school year, my district is asking me to perform a number of evaluations on instructional programs. This is where my question/concern comes in. You mention that hypothesis testing is all about testing samples. However, I use all the students in my district when I make these comparisons. Since I am using the entire “population” in my evaluations (I don’t select a sample of third grade students, for example, but I use all 700 third graders), am I somehow misusing the tests? Or can I rest assured that my district’s student population is only a sample of the universal population of students?

June 10, 2024 at 1:50 pm

I hope you are finding the book helpful!

Yes, the purpose of hypothesis testing is to infer the properties of a population while accounting for random sampling error.

In your case, it comes down to how you want to use the results. Who do you want the results to apply to?

If you’re summarizing the sample, looking for trends and patterns, or evaluating those students and don’t plan to apply those results to other students, you don’t need hypothesis testing because there is no sampling error. They are the population and you can just use descriptive statistics. In this case, you’d only need to focus on the practical significance of the effect sizes.

On the other hand, if you want to apply the results from this group to other students, you’ll need hypothesis testing. However, there is the complicating issue of what population your sample of students represent. I’m sure your district has its own unique characteristics, demographics, etc. Your district’s students probably don’t adequately represent a universal population. At the very least, you’d need to recognize any special attributes of your district and how they could bias the results when trying to apply them outside the district. Or they might apply to similar districts in your region.

However, I’d imagine your 3rd graders probably adequately represent future classes of 3rd graders in your district. You need to be alert to changing demographics. At least in the short run I’d imagine they’d be representative of future classes.

Think about how these results will be used. Do they just apply to the students you measured? Then you don’t need hypothesis tests. However, if the results are being used to infer things about other students outside of the sample, you’ll need hypothesis testing along with considering how well your students represent the other students and how they differ.

I hope that helps!

June 10, 2024 at 3:21 pm

Thank you so much, Jim, for the suggestions in terms of what I need to think about and consider! You are always so clear in your explanations!!!!

June 10, 2024 at 3:22 pm

You’re very welcome! Best of luck with your evaluations!

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Hypothesis Testing

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A hypothesis test is a statistical inference method used to test the significance of a proposed (hypothesized) relation between population statistics (parameters) and their corresponding sample estimators . In other words, hypothesis tests are used to determine if there is enough evidence in a sample to prove a hypothesis true for the entire population.

The test considers two hypotheses: the null hypothesis , which is a statement meant to be tested, usually something like "there is no effect" with the intention of proving this false, and the alternate hypothesis , which is the statement meant to stand after the test is performed. The two hypotheses must be mutually exclusive ; moreover, in most applications, the two are complementary (one being the negation of the other). The test works by comparing the \(p\)-value to the level of significance (a chosen target). If the \(p\)-value is less than or equal to the level of significance, then the null hypothesis is rejected.

When analyzing data, only samples of a certain size might be manageable as efficient computations. In some situations the error terms follow a continuous or infinite distribution, hence the use of samples to suggest accuracy of the chosen test statistics. The method of hypothesis testing gives an advantage over guessing what distribution or which parameters the data follows.

Definitions and Methodology

Hypothesis test and confidence intervals.

In statistical inference, properties (parameters) of a population are analyzed by sampling data sets. Given assumptions on the distribution, i.e. a statistical model of the data, certain hypotheses can be deduced from the known behavior of the model. These hypotheses must be tested against sampled data from the population.

The null hypothesis \((\)denoted \(H_0)\) is a statement that is assumed to be true. If the null hypothesis is rejected, then there is enough evidence (statistical significance) to accept the alternate hypothesis \((\)denoted \(H_1).\) Before doing any test for significance, both hypotheses must be clearly stated and non-conflictive, i.e. mutually exclusive, statements. Rejecting the null hypothesis, given that it is true, is called a type I error and it is denoted \(\alpha\), which is also its probability of occurrence. Failing to reject the null hypothesis, given that it is false, is called a type II error and it is denoted \(\beta\), which is also its probability of occurrence. Also, \(\alpha\) is known as the significance level , and \(1-\beta\) is known as the power of the test. \(H_0\) \(\textbf{is true}\)\(\hspace{15mm}\) \(H_0\) \(\textbf{is false}\) \(\textbf{Reject}\) \(H_0\)\(\hspace{10mm}\) Type I error Correct Decision \(\textbf{Reject}\) \(H_1\) Correct Decision Type II error The test statistic is the standardized value following the sampled data under the assumption that the null hypothesis is true, and a chosen particular test. These tests depend on the statistic to be studied and the assumed distribution it follows, e.g. the population mean following a normal distribution. The \(p\)-value is the probability of observing an extreme test statistic in the direction of the alternate hypothesis, given that the null hypothesis is true. The critical value is the value of the assumed distribution of the test statistic such that the probability of making a type I error is small.

Methodologies: Given an estimator \(\hat \theta\) of a population statistic \(\theta\), following a probability distribution \(P(T)\), computed from a sample \(\mathcal{S},\) and given a significance level \(\alpha\) and test statistic \(t^*,\) define \(H_0\) and \(H_1;\) compute the test statistic \(t^*.\) \(p\)-value Approach (most prevalent): Find the \(p\)-value using \(t^*\) (right-tailed). If the \(p\)-value is at most \(\alpha,\) reject \(H_0\). Otherwise, reject \(H_1\). Critical Value Approach: Find the critical value solving the equation \(P(T\geq t_\alpha)=\alpha\) (right-tailed). If \(t^*>t_\alpha\), reject \(H_0\). Otherwise, reject \(H_1\). Note: Failing to reject \(H_0\) only means inability to accept \(H_1\), and it does not mean to accept \(H_0\).

Assume a normally distributed population has recorded cholesterol levels with various statistics computed. From a sample of 100 subjects in the population, the sample mean was 214.12 mg/dL (milligrams per deciliter), with a sample standard deviation of 45.71 mg/dL. Perform a hypothesis test, with significance level 0.05, to test if there is enough evidence to conclude that the population mean is larger than 200 mg/dL. Hypothesis Test We will perform a hypothesis test using the \(p\)-value approach with significance level \(\alpha=0.05:\) Define \(H_0\): \(\mu=200\). Define \(H_1\): \(\mu>200\). Since our values are normally distributed, the test statistic is \(z^*=\frac{\bar X - \mu_0}{\frac{s}{\sqrt{n}}}=\frac{214.12 - 200}{\frac{45.71}{\sqrt{100}}}\approx 3.09\). Using a standard normal distribution, we find that our \(p\)-value is approximately \(0.001\). Since the \(p\)-value is at most \(\alpha=0.05,\) we reject \(H_0\). Therefore, we can conclude that the test shows sufficient evidence to support the claim that \(\mu\) is larger than \(200\) mg/dL.

If the sample size was smaller, the normal and \(t\)-distributions behave differently. Also, the question itself must be managed by a double-tail test instead.

Assume a population's cholesterol levels are recorded and various statistics are computed. From a sample of 25 subjects, the sample mean was 214.12 mg/dL (milligrams per deciliter), with a sample standard deviation of 45.71 mg/dL. Perform a hypothesis test, with significance level 0.05, to test if there is enough evidence to conclude that the population mean is not equal to 200 mg/dL. Hypothesis Test We will perform a hypothesis test using the \(p\)-value approach with significance level \(\alpha=0.05\) and the \(t\)-distribution with 24 degrees of freedom: Define \(H_0\): \(\mu=200\). Define \(H_1\): \(\mu\neq 200\). Using the \(t\)-distribution, the test statistic is \(t^*=\frac{\bar X - \mu_0}{\frac{s}{\sqrt{n}}}=\frac{214.12 - 200}{\frac{45.71}{\sqrt{25}}}\approx 1.54\). Using a \(t\)-distribution with 24 degrees of freedom, we find that our \(p\)-value is approximately \(2(0.068)=0.136\). We have multiplied by two since this is a two-tailed argument, i.e. the mean can be smaller than or larger than. Since the \(p\)-value is larger than \(\alpha=0.05,\) we fail to reject \(H_0\). Therefore, the test does not show sufficient evidence to support the claim that \(\mu\) is not equal to \(200\) mg/dL.

The complement of the rejection on a two-tailed hypothesis test (with significance level \(\alpha\)) for a population parameter \(\theta\) is equivalent to finding a confidence interval \((\)with confidence level \(1-\alpha)\) for the population parameter \(\theta\). If the assumption on the parameter \(\theta\) falls inside the confidence interval, then the test has failed to reject the null hypothesis \((\)with \(p\)-value greater than \(\alpha).\) Otherwise, if \(\theta\) does not fall in the confidence interval, then the null hypothesis is rejected in favor of the alternate \((\)with \(p\)-value at most \(\alpha).\)

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Hypothesis Testing – A Deep Dive into Hypothesis Testing, The Backbone of Statistical Inference

• September 21, 2023

Explore the intricacies of hypothesis testing, a cornerstone of statistical analysis. Dive into methods, interpretations, and applications for making data-driven decisions.

In this Blog post we will learn:

• What is Hypothesis Testing?
• Steps in Hypothesis Testing 2.1. Set up Hypotheses: Null and Alternative 2.2. Choose a Significance Level (α) 2.3. Calculate a test statistic and P-Value 2.4. Make a Decision
• Example : Testing a new drug.
• Example in python

1. What is Hypothesis Testing?

In simple terms, hypothesis testing is a method used to make decisions or inferences about population parameters based on sample data. Imagine being handed a dice and asked if it’s biased. By rolling it a few times and analyzing the outcomes, you’d be engaging in the essence of hypothesis testing.

Think of hypothesis testing as the scientific method of the statistics world. Suppose you hear claims like “This new drug works wonders!” or “Our new website design boosts sales.” How do you know if these statements hold water? Enter hypothesis testing.

2. Steps in Hypothesis Testing

• Set up Hypotheses : Begin with a null hypothesis (H0) and an alternative hypothesis (Ha).
• Choose a Significance Level (α) : Typically 0.05, this is the probability of rejecting the null hypothesis when it’s actually true. Think of it as the chance of accusing an innocent person.
• Calculate Test statistic and P-Value : Gather evidence (data) and calculate a test statistic.
• p-value : This is the probability of observing the data, given that the null hypothesis is true. A small p-value (typically ≤ 0.05) suggests the data is inconsistent with the null hypothesis.
• Decision Rule : If the p-value is less than or equal to α, you reject the null hypothesis in favor of the alternative.

2.1. Set up Hypotheses: Null and Alternative

Before diving into testing, we must formulate hypotheses. The null hypothesis (H0) represents the default assumption, while the alternative hypothesis (H1) challenges it.

For instance, in drug testing, H0 : “The new drug is no better than the existing one,” H1 : “The new drug is superior .”

2.2. Choose a Significance Level (α)

When You collect and analyze data to test H0 and H1 hypotheses. Based on your analysis, you decide whether to reject the null hypothesis in favor of the alternative, or fail to reject / Accept the null hypothesis.

The significance level, often denoted by $α$, represents the probability of rejecting the null hypothesis when it is actually true.

In other words, it’s the risk you’re willing to take of making a Type I error (false positive).

Type I Error (False Positive) :

• Symbolized by the Greek letter alpha (α).
• Occurs when you incorrectly reject a true null hypothesis . In other words, you conclude that there is an effect or difference when, in reality, there isn’t.
• The probability of making a Type I error is denoted by the significance level of a test. Commonly, tests are conducted at the 0.05 significance level , which means there’s a 5% chance of making a Type I error .
• Commonly used significance levels are 0.01, 0.05, and 0.10, but the choice depends on the context of the study and the level of risk one is willing to accept.

Example : If a drug is not effective (truth), but a clinical trial incorrectly concludes that it is effective (based on the sample data), then a Type I error has occurred.

Type II Error (False Negative) :

• Symbolized by the Greek letter beta (β).
• Occurs when you accept a false null hypothesis . This means you conclude there is no effect or difference when, in reality, there is.
• The probability of making a Type II error is denoted by β. The power of a test (1 – β) represents the probability of correctly rejecting a false null hypothesis.

Example : If a drug is effective (truth), but a clinical trial incorrectly concludes that it is not effective (based on the sample data), then a Type II error has occurred.

Balancing the Errors :

In practice, there’s a trade-off between Type I and Type II errors. Reducing the risk of one typically increases the risk of the other. For example, if you want to decrease the probability of a Type I error (by setting a lower significance level), you might increase the probability of a Type II error unless you compensate by collecting more data or making other adjustments.

It’s essential to understand the consequences of both types of errors in any given context. In some situations, a Type I error might be more severe, while in others, a Type II error might be of greater concern. This understanding guides researchers in designing their experiments and choosing appropriate significance levels.

2.3. Calculate a test statistic and P-Value

Test statistic : A test statistic is a single number that helps us understand how far our sample data is from what we’d expect under a null hypothesis (a basic assumption we’re trying to test against). Generally, the larger the test statistic, the more evidence we have against our null hypothesis. It helps us decide whether the differences we observe in our data are due to random chance or if there’s an actual effect.

P-value : The P-value tells us how likely we would get our observed results (or something more extreme) if the null hypothesis were true. It’s a value between 0 and 1. – A smaller P-value (typically below 0.05) means that the observation is rare under the null hypothesis, so we might reject the null hypothesis. – A larger P-value suggests that what we observed could easily happen by random chance, so we might not reject the null hypothesis.

2.4. Make a Decision

Relationship between $α$ and P-Value

When conducting a hypothesis test:

We then calculate the p-value from our sample data and the test statistic.

Finally, we compare the p-value to our chosen $α$:

• If $p−value≤α$: We reject the null hypothesis in favor of the alternative hypothesis. The result is said to be statistically significant.
• If $p−value>α$: We fail to reject the null hypothesis. There isn’t enough statistical evidence to support the alternative hypothesis.

3. Example : Testing a new drug.

Imagine we are investigating whether a new drug is effective at treating headaches faster than drug B.

Setting Up the Experiment : You gather 100 people who suffer from headaches. Half of them (50 people) are given the new drug (let’s call this the ‘Drug Group’), and the other half are given a sugar pill, which doesn’t contain any medication.

• Set up Hypotheses : Before starting, you make a prediction:
• Null Hypothesis (H0): The new drug has no effect. Any difference in healing time between the two groups is just due to random chance.
• Alternative Hypothesis (H1): The new drug does have an effect. The difference in healing time between the two groups is significant and not just by chance.

Calculate Test statistic and P-Value : After the experiment, you analyze the data. The “test statistic” is a number that helps you understand the difference between the two groups in terms of standard units.

For instance, let’s say:

• The average healing time in the Drug Group is 2 hours.
• The average healing time in the Placebo Group is 3 hours.

The test statistic helps you understand how significant this 1-hour difference is. If the groups are large and the spread of healing times in each group is small, then this difference might be significant. But if there’s a huge variation in healing times, the 1-hour difference might not be so special.

Imagine the P-value as answering this question: “If the new drug had NO real effect, what’s the probability that I’d see a difference as extreme (or more extreme) as the one I found, just by random chance?”

For instance:

• P-value of 0.01 means there’s a 1% chance that the observed difference (or a more extreme difference) would occur if the drug had no effect. That’s pretty rare, so we might consider the drug effective.
• P-value of 0.5 means there’s a 50% chance you’d see this difference just by chance. That’s pretty high, so we might not be convinced the drug is doing much.
• If the P-value is less than ($α$) 0.05: the results are “statistically significant,” and they might reject the null hypothesis , believing the new drug has an effect.
• If the P-value is greater than ($α$) 0.05: the results are not statistically significant, and they don’t reject the null hypothesis , remaining unsure if the drug has a genuine effect.

4. Example in python

For simplicity, let’s say we’re using a t-test (common for comparing means). Let’s dive into Python:

Making a Decision : “The results are statistically significant! p-value < 0.05 , The drug seems to have an effect!” If not, we’d say, “Looks like the drug isn’t as miraculous as we thought.”

5. Conclusion

Hypothesis testing is an indispensable tool in data science, allowing us to make data-driven decisions with confidence. By understanding its principles, conducting tests properly, and considering real-world applications, you can harness the power of hypothesis testing to unlock valuable insights from your data.

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Correlation – connecting the dots, the role of correlation in data analysis, sampling and sampling distributions – a comprehensive guide on sampling and sampling distributions, law of large numbers – a deep dive into the world of statistics, central limit theorem – a deep dive into central limit theorem and its significance in statistics, skewness and kurtosis – peaks and tails, understanding data through skewness and kurtosis”, similar articles, complete introduction to linear regression in r, how to implement common statistical significance tests and find the p value, logistic regression – a complete tutorial with examples in r.

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Hypothesis testing.

Key Topics:

• Basic approach
• Null and alternative hypothesis
• Decision making and the p -value
• Z-test & Nonparametric alternative

Basic approach to hypothesis testing

• State a model describing the relationship between the explanatory variables and the outcome variable(s) in the population and the nature of the variability. State all of your assumptions .
• Specify the null and alternative hypotheses in terms of the parameters of the model.
• Invent a test statistic that will tend to be different under the null and alternative hypotheses.
• Using the assumptions of step 1, find the theoretical sampling distribution of the statistic under the null hypothesis of step 2. Ideally the form of the sampling distribution should be one of the “standard distributions”(e.g. normal, t , binomial..)
• Calculate a p -value , as the area under the sampling distribution more extreme than your statistic. Depends on the form of the alternative hypothesis.
• Choose your acceptable type 1 error rate (alpha) and apply the decision rule : reject the null hypothesis if the p-value is less than alpha, otherwise do not reject.

• \(\frac{\bar{X}-\mu_0}{\sigma / \sqrt{n}}\)
• general form is: (estimate - value we are testing)/(st.dev of the estimate)
• z-statistic follows N(0,1) distribution
• 2 × the area above |z|, area above z,or area below z, or
• compare the statistic to a critical value, |z| ≥ z α/2 , z ≥ z α , or z ≤ - z α
• Choose the acceptable level of Alpha = 0.05, we conclude …. ?

Making the Decision

It is either likely or unlikely that we would collect the evidence we did given the initial assumption. (Note: “likely” or “unlikely” is measured by calculating a probability!)

If it is likely , then we “ do not reject ” our initial assumption. There is not enough evidence to do otherwise.

If it is unlikely , then:

• either our initial assumption is correct and we experienced an unusual event or,
• our initial assumption is incorrect

In statistics, if it is unlikely, we decide to “ reject ” our initial assumption.

Example: Criminal Trial Analogy

First, state 2 hypotheses, the null hypothesis (“H 0 ”) and the alternative hypothesis (“H A ”)

• H 0 : Defendant is not guilty.
• H A : Defendant is guilty.

Usually the H 0 is a statement of “no effect”, or “no change”, or “chance only” about a population parameter.

While the H A , depending on the situation, is that there is a difference, trend, effect, or a relationship with respect to a population parameter.

• It can one-sided and two-sided.
• In two-sided we only care there is a difference, but not the direction of it. In one-sided we care about a particular direction of the relationship. We want to know if the value is strictly larger or smaller.

Then, collect evidence, such as finger prints, blood spots, hair samples, carpet fibers, shoe prints, ransom notes, handwriting samples, etc. (In statistics, the data are the evidence.)

Next, you make your initial assumption.

• Defendant is innocent until proven guilty.

In statistics, we always assume the null hypothesis is true .

Then, make a decision based on the available evidence.

• If there is sufficient evidence (“beyond a reasonable doubt”), reject the null hypothesis . (Behave as if defendant is guilty.)
• If there is not enough evidence, do not reject the null hypothesis . (Behave as if defendant is not guilty.)

If the observed outcome, e.g., a sample statistic, is surprising under the assumption that the null hypothesis is true, but more probable if the alternative is true, then this outcome is evidence against H 0 and in favor of H A .

An observed effect so large that it would rarely occur by chance is called statistically significant (i.e., not likely to happen by chance).

Using the p -value to make the decision

The p -value represents how likely we would be to observe such an extreme sample if the null hypothesis were true. The p -value is a probability computed assuming the null hypothesis is true, that the test statistic would take a value as extreme or more extreme than that actually observed. Since it's a probability, it is a number between 0 and 1. The closer the number is to 0 means the event is “unlikely.” So if p -value is “small,” (typically, less than 0.05), we can then reject the null hypothesis.

Significance level and p -value

Significance level, α, is a decisive value for p -value. In this context, significant does not mean “important”, but it means “not likely to happened just by chance”.

α is the maximum probability of rejecting the null hypothesis when the null hypothesis is true. If α = 1 we always reject the null, if α = 0 we never reject the null hypothesis. In articles, journals, etc… you may read: “The results were significant ( p <0.05).” So if p =0.03, it's significant at the level of α = 0.05 but not at the level of α = 0.01. If we reject the H 0 at the level of α = 0.05 (which corresponds to 95% CI), we are saying that if H 0 is true, the observed phenomenon would happen no more than 5% of the time (that is 1 in 20). If we choose to compare the p -value to α = 0.01, we are insisting on a stronger evidence!

So, what kind of error could we make? No matter what decision we make, there is always a chance we made an error.

Errors in Criminal Trial:

Errors in Hypothesis Testing

Type I error (False positive): The null hypothesis is rejected when it is true.

• α is the maximum probability of making a Type I error.

Type II error (False negative): The null hypothesis is not rejected when it is false.

• β is the probability of making a Type II error

There is always a chance of making one of these errors. But, a good scientific study will minimize the chance of doing so!

The power of a statistical test is its probability of rejecting the null hypothesis if the null hypothesis is false. That is, power is the ability to correctly reject H 0 and detect a significant effect. In other words, power is one minus the type II error risk.

\(\text{Power }=1-\beta = P\left(\text{reject} H_0 | H_0 \text{is false } \right)\)

Which error is worse?

Type I = you are innocent, yet accused of cheating on the test. Type II = you cheated on the test, but you are found innocent.

This depends on the context of the problem too. But in most cases scientists are trying to be “conservative”; it's worse to make a spurious discovery than to fail to make a good one. Our goal it to increase the power of the test that is to minimize the length of the CI.

We need to keep in mind:

• the effect of the sample size,
• the correctness of the underlying assumptions about the population,
• statistical vs. practical significance, etc…

(see the handout). To study the tradeoffs between the sample size, α, and Type II error we can use power and operating characteristic curves.

What type of error might we have made?

Type I error is claiming that average student height is not 65 inches, when it really is. Type II error is failing to claim that the average student height is not 65in when it is.

We rejected the null hypothesis, i.e., claimed that the height is not 65, thus making potentially a Type I error. But sometimes the p -value is too low because of the large sample size, and we may have statistical significance but not really practical significance! That's why most statisticians are much more comfortable with using CI than tests.

There is a need for a further generalization. What if we can't assume that σ is known? In this case we would use s (the sample standard deviation) to estimate σ.

If the sample is very large, we can treat σ as known by assuming that σ = s . According to the law of large numbers, this is not too bad a thing to do. But if the sample is small, the fact that we have to estimate both the standard deviation and the mean adds extra uncertainty to our inference. In practice this means that we need a larger multiplier for the standard error.

We need one-sample t -test.

One sample t -test

• Assume data are independently sampled from a normal distribution with unknown mean μ and variance σ 2 . Make an initial assumption, μ 0 .

• t-statistic: \(\frac{\bar{X}-\mu_0}{s / \sqrt{n}}\) where s is a sample st.dev.
• t-statistic follows t -distribution with df = n - 1
• Alpha = 0.05, we conclude ….

Testing for the population proportion

Let's go back to our CNN poll. Assume we have a SRS of 1,017 adults.

We are interested in testing the following hypothesis: H 0 : p = 0.50 vs. p > 0.50

What is the test statistic?

If alpha = 0.05, what do we conclude?

We will see more details in the next lesson on proportions, then distributions, and possible tests.

• How it works

Hypothesis Testing – A Complete Guide with Examples

Published by Alvin Nicolas at August 14th, 2021 , Revised On October 26, 2023

In statistics, hypothesis testing is a critical tool. It allows us to make informed decisions about populations based on sample data. Whether you are a researcher trying to prove a scientific point, a marketer analysing A/B test results, or a manufacturer ensuring quality control, hypothesis testing plays a pivotal role. This guide aims to introduce you to the concept and walk you through real-world examples.

What is a Hypothesis and a Hypothesis Testing?

A hypothesis is considered a belief or assumption that has to be accepted, rejected, proved or disproved. In contrast, a research hypothesis is a research question for a researcher that has to be proven correct or incorrect through investigation.

What is Hypothesis Testing?

Hypothesis testing  is a scientific method used for making a decision and drawing conclusions by using a statistical approach. It is used to suggest new ideas by testing theories to know whether or not the sample data supports research. A research hypothesis is a predictive statement that has to be tested using scientific methods that join an independent variable to a dependent variable.  

Example: The academic performance of student A is better than student B

Characteristics of the Hypothesis to be Tested

A hypothesis should be:

• Clear and precise
• Capable of being tested
• Able to relate to a variable
• Stated in simple terms
• Consistent with known facts
• Limited in scope and specific
• Tested in a limited timeframe
• Explain the facts in detail

What is a Null Hypothesis and Alternative Hypothesis?

A  null hypothesis  is a hypothesis when there is no significant relationship between the dependent and the participants’ independent  variables . 

In simple words, it’s a hypothesis that has been put forth but hasn’t been proved as yet. A researcher aims to disprove the theory. The abbreviation “Ho” is used to denote a null hypothesis.

If you want to compare two methods and assume that both methods are equally good, this assumption is considered the null hypothesis.

Example: In an automobile trial, you feel that the new vehicle’s mileage is similar to the previous model of the car, on average. You can write it as: Ho: there is no difference between the mileage of both vehicles. If your findings don’t support your hypothesis and you get opposite results, this outcome will be considered an alternative hypothesis.

If you assume that one method is better than another method, then it’s considered an alternative hypothesis. The alternative hypothesis is the theory that a researcher seeks to prove and is typically denoted by H1 or HA.

If you support a null hypothesis, it means you’re not supporting the alternative hypothesis. Similarly, if you reject a null hypothesis, it means you are recommending the alternative hypothesis.

Example: In an automobile trial, you feel that the new vehicle’s mileage is better than the previous model of the vehicle. You can write it as; Ha: the two vehicles have different mileage. On average/ the fuel consumption of the new vehicle model is better than the previous model.

If a null hypothesis is rejected during the hypothesis test, even if it’s true, then it is considered as a type-I error. On the other hand, if you don’t dismiss a hypothesis, even if it’s false because you could not identify its falseness, it’s considered a type-II error.

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How to Conduct Hypothesis Testing?

Here is a step-by-step guide on how to conduct hypothesis testing.

Step 1: State the Null and Alternative Hypothesis

Once you develop a research hypothesis, it’s important to state it is as a Null hypothesis (Ho) and an Alternative hypothesis (Ha) to test it statistically.

A null hypothesis is a preferred choice as it provides the opportunity to test the theory. In contrast, you can accept the alternative hypothesis when the null hypothesis has been rejected.

Example: You want to identify a relationship between obesity of men and women and the modern living style. You develop a hypothesis that women, on average, gain weight quickly compared to men. Then you write it as: Ho: Women, on average, don’t gain weight quickly compared to men. Ha: Women, on average, gain weight quickly compared to men.

Step 2: Data Collection

Hypothesis testing follows the statistical method, and statistics are all about data. It’s challenging to gather complete information about a specific population you want to study. You need to  gather the data  obtained through a large number of samples from a specific population. 

Example: Suppose you want to test the difference in the rate of obesity between men and women. You should include an equal number of men and women in your sample. Then investigate various aspects such as their lifestyle, eating patterns and profession, and any other variables that may influence average weight. You should also determine your study’s scope, whether it applies to a specific group of population or worldwide population. You can use available information from various places, countries, and regions.

Step 3: Select Appropriate Statistical Test

There are many  types of statistical tests , but we discuss the most two common types below, such as One-sided and two-sided tests.

Note: Your choice of the type of test depends on the purpose of your study 

One-sided Test

In the one-sided test, the values of rejecting a null hypothesis are located in one tail of the probability distribution. The set of values is less or higher than the critical value of the test. It is also called a one-tailed test of significance.

Example: If you want to test that all mangoes in a basket are ripe. You can write it as: Ho: All mangoes in the basket, on average, are ripe. If you find all ripe mangoes in the basket, the null hypothesis you developed will be true.

Two-sided Test

In the two-sided test, the values of rejecting a null hypothesis are located on both tails of the probability distribution. The set of values is less or higher than the first critical value of the test and higher than the second critical value test. It is also called a two-tailed test of significance. 

Example: Nothing can be explicitly said whether all mangoes are ripe in the basket. If you reject the null hypothesis (Ho: All mangoes in the basket, on average, are ripe), then it means all mangoes in the basket are not likely to be ripe. A few mangoes could be raw as well.

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Step 4: Select the Level of Significance

When you reject a null hypothesis, even if it’s true during a statistical hypothesis, it is considered the  significance level . It is the probability of a type one error. The significance should be as minimum as possible to avoid the type-I error, which is considered severe and should be avoided. 

If the significance level is minimum, then it prevents the researchers from false claims. 

The significance level is denoted by  P,  and it has given the value of 0.05 (P=0.05)

If the P-Value is less than 0.05, then the difference will be significant. If the P-value is higher than 0.05, then the difference is non-significant.

Example: Suppose you apply a one-sided test to test whether women gain weight quickly compared to men. You get to know about the average weight between men and women and the factors promoting weight gain.

Step 5: Find out Whether the Null Hypothesis is Rejected or Supported

After conducting a statistical test, you should identify whether your null hypothesis is rejected or accepted based on the test results. It would help if you observed the P-value for this.

Example: If you find the P-value of your test is less than 0.5/5%, then you need to reject your null hypothesis (Ho: Women, on average, don’t gain weight quickly compared to men). On the other hand, if a null hypothesis is rejected, then it means the alternative hypothesis might be true (Ha: Women, on average, gain weight quickly compared to men. If you find your test’s P-value is above 0.5/5%, then it means your null hypothesis is true.

Step 6: Present the Outcomes of your Study

The final step is to present the  outcomes of your study . You need to ensure whether you have met the objectives of your research or not. 

In the discussion section and  conclusion , you can present your findings by using supporting evidence and conclude whether your null hypothesis was rejected or supported.

In the result section, you can summarise your study’s outcomes, including the average difference and P-value of the two groups.

If we talk about the findings, our study your results will be as follows:

Example: In the study of identifying whether women gain weight quickly compared to men, we found the P-value is less than 0.5. Hence, we can reject the null hypothesis (Ho: Women, on average, don’t gain weight quickly than men) and conclude that women may likely gain weight quickly than men.

Did you know in your academic paper you should not mention whether you have accepted or rejected the null hypothesis? 

Always remember that you either conclude to reject Ho in favor of Haor   do not reject Ho . It would help if you never rejected  Ha  or even  accept Ha .

Suppose your null hypothesis is rejected in the hypothesis testing. If you conclude  reject Ho in favor of Haor   do not reject Ho,  then it doesn’t mean that the null hypothesis is true. It only means that there is a lack of evidence against Ho in favour of Ha. If your null hypothesis is not true, then the alternative hypothesis is likely to be true.

Example: We found that the P-value is less than 0.5. Hence, we can conclude reject Ho in favour of Ha (Ho: Women, on average, don’t gain weight quickly than men) reject Ho in favour of Ha. However, rejected in favour of Ha means (Ha: women may likely to gain weight quickly than men)

Frequently Asked Questions

What are the 3 types of hypothesis test.

The 3 types of hypothesis tests are:

• One-Sample Test : Compare sample data to a known population value.
• Two-Sample Test : Compare means between two sample groups.
• ANOVA : Analyze variance among multiple groups to determine significant differences.

What is a hypothesis?

A hypothesis is a proposed explanation or prediction about a phenomenon, often based on observations. It serves as a starting point for research or experimentation, providing a testable statement that can either be supported or refuted through data and analysis. In essence, it’s an educated guess that drives scientific inquiry.

What are null hypothesis?

A null hypothesis (often denoted as H0) suggests that there is no effect or difference in a study or experiment. It represents a default position or status quo. Statistical tests evaluate data to determine if there’s enough evidence to reject this null hypothesis.

What is the probability value?

The probability value, or p-value, is a measure used in statistics to determine the significance of an observed effect. It indicates the probability of obtaining the observed results, or more extreme, if the null hypothesis were true. A small p-value (typically <0.05) suggests evidence against the null hypothesis, warranting its rejection.

What is p value?

The p-value is a fundamental concept in statistical hypothesis testing. It represents the probability of observing a test statistic as extreme, or more so, than the one calculated from sample data, assuming the null hypothesis is true. A low p-value suggests evidence against the null, possibly justifying its rejection.

What is a t test?

A t-test is a statistical test used to compare the means of two groups. It determines if observed differences between the groups are statistically significant or if they likely occurred by chance. Commonly applied in research, there are different t-tests, including independent, paired, and one-sample, tailored to various data scenarios.

When to reject null hypothesis?

Reject the null hypothesis when the test statistic falls into a predefined rejection region or when the p-value is less than the chosen significance level (commonly 0.05). This suggests that the observed data is unlikely under the null hypothesis, indicating evidence for the alternative hypothesis. Always consider the study’s context.

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What Is Hypothesis Testing?

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4 Step Process

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• Fundamental Analysis

Hypothesis Testing: 4 Steps and Example

Hypothesis testing, sometimes called significance testing, is an act in statistics whereby an analyst tests an assumption regarding a population parameter. The methodology employed by the analyst depends on the nature of the data used and the reason for the analysis.

Hypothesis testing is used to assess the plausibility of a hypothesis by using sample data. Such data may come from a larger population or a data-generating process. The word "population" will be used for both of these cases in the following descriptions.

Key Takeaways

• Hypothesis testing is used to assess the plausibility of a hypothesis by using sample data.
• The test provides evidence concerning the plausibility of the hypothesis, given the data.
• Statistical analysts test a hypothesis by measuring and examining a random sample of the population being analyzed.
• The four steps of hypothesis testing include stating the hypotheses, formulating an analysis plan, analyzing the sample data, and analyzing the result.

How Hypothesis Testing Works

In hypothesis testing, an  analyst  tests a statistical sample, intending to provide evidence on the plausibility of the null hypothesis. Statistical analysts measure and examine a random sample of the population being analyzed. All analysts use a random population sample to test two different hypotheses: the null hypothesis and the alternative hypothesis.

The null hypothesis is usually a hypothesis of equality between population parameters; e.g., a null hypothesis may state that the population mean return is equal to zero. The alternative hypothesis is effectively the opposite of a null hypothesis. Thus, they are mutually exclusive , and only one can be true. However, one of the two hypotheses will always be true.

The null hypothesis is a statement about a population parameter, such as the population mean, that is assumed to be true.

• State the hypotheses.
• Formulate an analysis plan, which outlines how the data will be evaluated.
• Carry out the plan and analyze the sample data.
• Analyze the results and either reject the null hypothesis, or state that the null hypothesis is plausible, given the data.

Example of Hypothesis Testing

If an individual wants to test that a penny has exactly a 50% chance of landing on heads, the null hypothesis would be that 50% is correct, and the alternative hypothesis would be that 50% is not correct. Mathematically, the null hypothesis is represented as Ho: P = 0.5. The alternative hypothesis is shown as "Ha" and is identical to the null hypothesis, except with the equal sign struck-through, meaning that it does not equal 50%.

A random sample of 100 coin flips is taken, and the null hypothesis is tested. If it is found that the 100 coin flips were distributed as 40 heads and 60 tails, the analyst would assume that a penny does not have a 50% chance of landing on heads and would reject the null hypothesis and accept the alternative hypothesis.

If there were 48 heads and 52 tails, then it is plausible that the coin could be fair and still produce such a result. In cases such as this where the null hypothesis is "accepted," the analyst states that the difference between the expected results (50 heads and 50 tails) and the observed results (48 heads and 52 tails) is "explainable by chance alone."

When Did Hypothesis Testing Begin?

Some statisticians attribute the first hypothesis tests to satirical writer John Arbuthnot in 1710, who studied male and female births in England after observing that in nearly every year, male births exceeded female births by a slight proportion. Arbuthnot calculated that the probability of this happening by chance was small, and therefore it was due to “divine providence.”

What are the Benefits of Hypothesis Testing?

Hypothesis testing helps assess the accuracy of new ideas or theories by testing them against data. This allows researchers to determine whether the evidence supports their hypothesis, helping to avoid false claims and conclusions. Hypothesis testing also provides a framework for decision-making based on data rather than personal opinions or biases. By relying on statistical analysis, hypothesis testing helps to reduce the effects of chance and confounding variables, providing a robust framework for making informed conclusions.

What are the Limitations of Hypothesis Testing?

Hypothesis testing relies exclusively on data and doesn’t provide a comprehensive understanding of the subject being studied. Additionally, the accuracy of the results depends on the quality of the available data and the statistical methods used. Inaccurate data or inappropriate hypothesis formulation may lead to incorrect conclusions or failed tests. Hypothesis testing can also lead to errors, such as analysts either accepting or rejecting a null hypothesis when they shouldn’t have. These errors may result in false conclusions or missed opportunities to identify significant patterns or relationships in the data.

Hypothesis testing refers to a statistical process that helps researchers determine the reliability of a study. By using a well-formulated hypothesis and set of statistical tests, individuals or businesses can make inferences about the population that they are studying and draw conclusions based on the data presented. All hypothesis testing methods have the same four-step process, which includes stating the hypotheses, formulating an analysis plan, analyzing the sample data, and analyzing the result.

Sage. " Introduction to Hypothesis Testing ," Page 4.

Elder Research. " Who Invented the Null Hypothesis? "

Formplus. " Hypothesis Testing: Definition, Uses, Limitations and Examples ."

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Teach yourself statistics

What is Hypothesis Testing?

A statistical hypothesis is an assumption about a population parameter . This assumption may or may not be true. Hypothesis testing refers to the formal procedures used by statisticians to accept or reject statistical hypotheses.

Statistical Hypotheses

The best way to determine whether a statistical hypothesis is true would be to examine the entire population. Since that is often impractical, researchers typically examine a random sample from the population. If sample data are not consistent with the statistical hypothesis, the hypothesis is rejected.

There are two types of statistical hypotheses.

• Null hypothesis . The null hypothesis, denoted by H o , is usually the hypothesis that sample observations result purely from chance.
• Alternative hypothesis . The alternative hypothesis, denoted by H 1 or H a , is the hypothesis that sample observations are influenced by some non-random cause.

For example, suppose we wanted to determine whether a coin was fair and balanced. A null hypothesis might be that half the flips would result in Heads and half, in Tails. The alternative hypothesis might be that the number of Heads and Tails would be very different. Symbolically, these hypotheses would be expressed as

H o : P = 0.5 H a : P ≠ 0.5

Suppose we flipped the coin 50 times, resulting in 40 Heads and 10 Tails. Given this result, we would be inclined to reject the null hypothesis. We would conclude, based on the evidence, that the coin was probably not fair and balanced.

Can We Accept the Null Hypothesis?

Some researchers say that a hypothesis test can have one of two outcomes: you accept the null hypothesis or you reject the null hypothesis. Many statisticians, however, take issue with the notion of "accepting the null hypothesis." Instead, they say: you reject the null hypothesis or you fail to reject the null hypothesis.

Why the distinction between "acceptance" and "failure to reject?" Acceptance implies that the null hypothesis is true. Failure to reject implies that the data are not sufficiently persuasive for us to prefer the alternative hypothesis over the null hypothesis.

Hypothesis Tests

Statisticians follow a formal process to determine whether to reject a null hypothesis, based on sample data. This process, called hypothesis testing , consists of four steps.

• State the hypotheses. This involves stating the null and alternative hypotheses. The hypotheses are stated in such a way that they are mutually exclusive. That is, if one is true, the other must be false.
• Formulate an analysis plan. The analysis plan describes how to use sample data to evaluate the null hypothesis. The evaluation often focuses around a single test statistic.
• Analyze sample data. Find the value of the test statistic (mean score, proportion, t statistic, z-score, etc.) described in the analysis plan.
• Interpret results. Apply the decision rule described in the analysis plan. If the value of the test statistic is unlikely, based on the null hypothesis, reject the null hypothesis.

Decision Errors

Two types of errors can result from a hypothesis test.

• Type I error . A Type I error occurs when the researcher rejects a null hypothesis when it is true. The probability of committing a Type I error is called the significance level . This probability is also called alpha , and is often denoted by α.
• Type II error . A Type II error occurs when the researcher fails to reject a null hypothesis that is false. The probability of committing a Type II error is called Beta , and is often denoted by β. The probability of not committing a Type II error is called the Power of the test.

Decision Rules

The analysis plan for a hypothesis test must include decision rules for rejecting the null hypothesis. In practice, statisticians describe these decision rules in two ways - with reference to a P-value or with reference to a region of acceptance.

• P-value. The strength of evidence in support of a null hypothesis is measured by the P-value . Suppose the test statistic is equal to S . The P-value is the probability of observing a test statistic as extreme as S , assuming the null hypothesis is true. If the P-value is less than the significance level, we reject the null hypothesis.

The set of values outside the region of acceptance is called the region of rejection . If the test statistic falls within the region of rejection, the null hypothesis is rejected. In such cases, we say that the hypothesis has been rejected at the α level of significance.

These approaches are equivalent. Some statistics texts use the P-value approach; others use the region of acceptance approach.

One-Tailed and Two-Tailed Tests

A test of a statistical hypothesis, where the region of rejection is on only one side of the sampling distribution , is called a one-tailed test . For example, suppose the null hypothesis states that the mean is less than or equal to 10. The alternative hypothesis would be that the mean is greater than 10. The region of rejection would consist of a range of numbers located on the right side of sampling distribution; that is, a set of numbers greater than 10.

A test of a statistical hypothesis, where the region of rejection is on both sides of the sampling distribution, is called a two-tailed test . For example, suppose the null hypothesis states that the mean is equal to 10. The alternative hypothesis would be that the mean is less than 10 or greater than 10. The region of rejection would consist of a range of numbers located on both sides of sampling distribution; that is, the region of rejection would consist partly of numbers that were less than 10 and partly of numbers that were greater than 10.

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A Beginner’s Guide to Hypothesis Testing in Business

• 30 Mar 2021

Becoming a more data-driven decision-maker can bring several benefits to your organization, enabling you to identify new opportunities to pursue and threats to abate. Rather than allowing subjective thinking to guide your business strategy, backing your decisions with data can empower your company to become more innovative and, ultimately, profitable.

If you’re new to data-driven decision-making, you might be wondering how data translates into business strategy. The answer lies in generating a hypothesis and verifying or rejecting it based on what various forms of data tell you.

Below is a look at hypothesis testing and the role it plays in helping businesses become more data-driven.

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What Is Hypothesis Testing?

To understand what hypothesis testing is, it’s important first to understand what a hypothesis is.

A hypothesis or hypothesis statement seeks to explain why something has happened, or what might happen, under certain conditions. It can also be used to understand how different variables relate to each other. Hypotheses are often written as if-then statements; for example, “If this happens, then this will happen.”

Hypothesis testing , then, is a statistical means of testing an assumption stated in a hypothesis. While the specific methodology leveraged depends on the nature of the hypothesis and data available, hypothesis testing typically uses sample data to extrapolate insights about a larger population.

Hypothesis Testing in Business

When it comes to data-driven decision-making, there’s a certain amount of risk that can mislead a professional. This could be due to flawed thinking or observations, incomplete or inaccurate data , or the presence of unknown variables. The danger in this is that, if major strategic decisions are made based on flawed insights, it can lead to wasted resources, missed opportunities, and catastrophic outcomes.

The real value of hypothesis testing in business is that it allows professionals to test their theories and assumptions before putting them into action. This essentially allows an organization to verify its analysis is correct before committing resources to implement a broader strategy.

As one example, consider a company that wishes to launch a new marketing campaign to revitalize sales during a slow period. Doing so could be an incredibly expensive endeavor, depending on the campaign’s size and complexity. The company, therefore, may wish to test the campaign on a smaller scale to understand how it will perform.

In this example, the hypothesis that’s being tested would fall along the lines of: “If the company launches a new marketing campaign, then it will translate into an increase in sales.” It may even be possible to quantify how much of a lift in sales the company expects to see from the effort. Pending the results of the pilot campaign, the business would then know whether it makes sense to roll it out more broadly.

Related: 9 Fundamental Data Science Skills for Business Professionals

Key Considerations for Hypothesis Testing

1. alternative hypothesis and null hypothesis.

In hypothesis testing, the hypothesis that’s being tested is known as the alternative hypothesis . Often, it’s expressed as a correlation or statistical relationship between variables. The null hypothesis , on the other hand, is a statement that’s meant to show there’s no statistical relationship between the variables being tested. It’s typically the exact opposite of whatever is stated in the alternative hypothesis.

For example, consider a company’s leadership team that historically and reliably sees $12 million in monthly revenue. They want to understand if reducing the price of their services will attract more customers and, in turn, increase revenue.

In this case, the alternative hypothesis may take the form of a statement such as: “If we reduce the price of our flagship service by five percent, then we’ll see an increase in sales and realize revenues greater than $12 million in the next month.”

The null hypothesis, on the other hand, would indicate that revenues wouldn’t increase from the base of $12 million, or might even decrease.

Check out the video below about the difference between an alternative and a null hypothesis, and subscribe to our YouTube channel for more explainer content.

2. Significance Level and P-Value

Statistically speaking, if you were to run the same scenario 100 times, you’d likely receive somewhat different results each time. If you were to plot these results in a distribution plot, you’d see the most likely outcome is at the tallest point in the graph, with less likely outcomes falling to the right and left of that point.

With this in mind, imagine you’ve completed your hypothesis test and have your results, which indicate there may be a correlation between the variables you were testing. To understand your results' significance, you’ll need to identify a p-value for the test, which helps note how confident you are in the test results.

In statistics, the p-value depicts the probability that, assuming the null hypothesis is correct, you might still observe results that are at least as extreme as the results of your hypothesis test. The smaller the p-value, the more likely the alternative hypothesis is correct, and the greater the significance of your results.

3. One-Sided vs. Two-Sided Testing

When it’s time to test your hypothesis, it’s important to leverage the correct testing method. The two most common hypothesis testing methods are one-sided and two-sided tests , or one-tailed and two-tailed tests, respectively.

Typically, you’d leverage a one-sided test when you have a strong conviction about the direction of change you expect to see due to your hypothesis test. You’d leverage a two-sided test when you’re less confident in the direction of change.

4. Sampling

To perform hypothesis testing in the first place, you need to collect a sample of data to be analyzed. Depending on the question you’re seeking to answer or investigate, you might collect samples through surveys, observational studies, or experiments.

A survey involves asking a series of questions to a random population sample and recording self-reported responses.

Observational studies involve a researcher observing a sample population and collecting data as it occurs naturally, without intervention.

Finally, an experiment involves dividing a sample into multiple groups, one of which acts as the control group. For each non-control group, the variable being studied is manipulated to determine how the data collected differs from that of the control group.

Learn How to Perform Hypothesis Testing

Hypothesis testing is a complex process involving different moving pieces that can allow an organization to effectively leverage its data and inform strategic decisions.

If you’re interested in better understanding hypothesis testing and the role it can play within your organization, one option is to complete a course that focuses on the process. Doing so can lay the statistical and analytical foundation you need to succeed.

Do you want to learn more about hypothesis testing? Explore Business Analytics —one of our online business essentials courses —and download our Beginner’s Guide to Data & Analytics .

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Hypothesis Testing (cont...)

Hypothesis testing, the null and alternative hypothesis.

In order to undertake hypothesis testing you need to express your research hypothesis as a null and alternative hypothesis. The null hypothesis and alternative hypothesis are statements regarding the differences or effects that occur in the population. You will use your sample to test which statement (i.e., the null hypothesis or alternative hypothesis) is most likely (although technically, you test the evidence against the null hypothesis). So, with respect to our teaching example, the null and alternative hypothesis will reflect statements about all statistics students on graduate management courses.

The null hypothesis is essentially the "devil's advocate" position. That is, it assumes that whatever you are trying to prove did not happen ( hint: it usually states that something equals zero). For example, the two different teaching methods did not result in different exam performances (i.e., zero difference). Another example might be that there is no relationship between anxiety and athletic performance (i.e., the slope is zero). The alternative hypothesis states the opposite and is usually the hypothesis you are trying to prove (e.g., the two different teaching methods did result in different exam performances). Initially, you can state these hypotheses in more general terms (e.g., using terms like "effect", "relationship", etc.), as shown below for the teaching methods example:

Depending on how you want to "summarize" the exam performances will determine how you might want to write a more specific null and alternative hypothesis. For example, you could compare the mean exam performance of each group (i.e., the "seminar" group and the "lectures-only" group). This is what we will demonstrate here, but other options include comparing the distributions , medians , amongst other things. As such, we can state:

Now that you have identified the null and alternative hypotheses, you need to find evidence and develop a strategy for declaring your "support" for either the null or alternative hypothesis. We can do this using some statistical theory and some arbitrary cut-off points. Both these issues are dealt with next.

Significance levels

The level of statistical significance is often expressed as the so-called p -value . Depending on the statistical test you have chosen, you will calculate a probability (i.e., the p -value) of observing your sample results (or more extreme) given that the null hypothesis is true . Another way of phrasing this is to consider the probability that a difference in a mean score (or other statistic) could have arisen based on the assumption that there really is no difference. Let us consider this statement with respect to our example where we are interested in the difference in mean exam performance between two different teaching methods. If there really is no difference between the two teaching methods in the population (i.e., given that the null hypothesis is true), how likely would it be to see a difference in the mean exam performance between the two teaching methods as large as (or larger than) that which has been observed in your sample?

So, you might get a p -value such as 0.03 (i.e., p = .03). This means that there is a 3% chance of finding a difference as large as (or larger than) the one in your study given that the null hypothesis is true. However, you want to know whether this is "statistically significant". Typically, if there was a 5% or less chance (5 times in 100 or less) that the difference in the mean exam performance between the two teaching methods (or whatever statistic you are using) is as different as observed given the null hypothesis is true, you would reject the null hypothesis and accept the alternative hypothesis. Alternately, if the chance was greater than 5% (5 times in 100 or more), you would fail to reject the null hypothesis and would not accept the alternative hypothesis. As such, in this example where p = .03, we would reject the null hypothesis and accept the alternative hypothesis. We reject it because at a significance level of 0.03 (i.e., less than a 5% chance), the result we obtained could happen too frequently for us to be confident that it was the two teaching methods that had an effect on exam performance.

Whilst there is relatively little justification why a significance level of 0.05 is used rather than 0.01 or 0.10, for example, it is widely used in academic research. However, if you want to be particularly confident in your results, you can set a more stringent level of 0.01 (a 1% chance or less; 1 in 100 chance or less).

One- and two-tailed predictions

When considering whether we reject the null hypothesis and accept the alternative hypothesis, we need to consider the direction of the alternative hypothesis statement. For example, the alternative hypothesis that was stated earlier is:

The alternative hypothesis tells us two things. First, what predictions did we make about the effect of the independent variable(s) on the dependent variable(s)? Second, what was the predicted direction of this effect? Let's use our example to highlight these two points.

Sarah predicted that her teaching method (independent variable: teaching method), whereby she not only required her students to attend lectures, but also seminars, would have a positive effect (that is, increased) students' performance (dependent variable: exam marks). If an alternative hypothesis has a direction (and this is how you want to test it), the hypothesis is one-tailed. That is, it predicts direction of the effect. If the alternative hypothesis has stated that the effect was expected to be negative, this is also a one-tailed hypothesis.

Alternatively, a two-tailed prediction means that we do not make a choice over the direction that the effect of the experiment takes. Rather, it simply implies that the effect could be negative or positive. If Sarah had made a two-tailed prediction, the alternative hypothesis might have been:

In other words, we simply take out the word "positive", which implies the direction of our effect. In our example, making a two-tailed prediction may seem strange. After all, it would be logical to expect that "extra" tuition (going to seminar classes as well as lectures) would either have a positive effect on students' performance or no effect at all, but certainly not a negative effect. However, this is just our opinion (and hope) and certainly does not mean that we will get the effect we expect. Generally speaking, making a one-tail prediction (i.e., and testing for it this way) is frowned upon as it usually reflects the hope of a researcher rather than any certainty that it will happen. Notable exceptions to this rule are when there is only one possible way in which a change could occur. This can happen, for example, when biological activity/presence in measured. That is, a protein might be "dormant" and the stimulus you are using can only possibly "wake it up" (i.e., it cannot possibly reduce the activity of a "dormant" protein). In addition, for some statistical tests, one-tailed tests are not possible.

Rejecting or failing to reject the null hypothesis

Let's return finally to the question of whether we reject or fail to reject the null hypothesis.

If our statistical analysis shows that the significance level is below the cut-off value we have set (e.g., either 0.05 or 0.01), we reject the null hypothesis and accept the alternative hypothesis. Alternatively, if the significance level is above the cut-off value, we fail to reject the null hypothesis and cannot accept the alternative hypothesis. You should note that you cannot accept the null hypothesis, but only find evidence against it.

4 Examples of Hypothesis Testing in Real Life

In statistics, hypothesis tests are used to test whether or not some hypothesis about a population parameter is true.

To perform a hypothesis test in the real world, researchers will obtain a random sample from the population and perform a hypothesis test on the sample data, using a null and alternative hypothesis:

• Null Hypothesis (H 0 ): The sample data occurs purely from chance.
• Alternative Hypothesis (H A ): The sample data is influenced by some non-random cause.

If the p-value of the hypothesis test is less than some significance level (e.g. α = .05), then we can reject the null hypothesis and conclude that we have sufficient evidence to say that the alternative hypothesis is true.

The following examples provide several situations where hypothesis tests are used in the real world.

Example 1: Biology

Hypothesis tests are often used in biology to determine whether some new treatment, fertilizer, pesticide, chemical, etc. causes increased growth, stamina, immunity, etc. in plants or animals.

For example, suppose a biologist believes that a certain fertilizer will cause plants to grow more during a one-month period than they normally do, which is currently 20 inches. To test this, she applies the fertilizer to each of the plants in her laboratory for one month.

She then performs a hypothesis test using the following hypotheses:

• H 0 : μ = 20 inches (the fertilizer will have no effect on the mean plant growth)
• H A : μ > 20 inches (the fertilizer will cause mean plant growth to increase)

If the p-value of the test is less than some significance level (e.g. α = .05), then she can reject the null hypothesis and conclude that the fertilizer leads to increased plant growth.

Example 2: Clinical Trials

Hypothesis tests are often used in clinical trials to determine whether some new treatment, drug, procedure, etc. causes improved outcomes in patients.

For example, suppose a doctor believes that a new drug is able to reduce blood pressure in obese patients. To test this, he may measure the blood pressure of 40 patients before and after using the new drug for one month.

He then performs a hypothesis test using the following hypotheses:

• H 0 : μ after = μ before (the mean blood pressure is the same before and after using the drug)
• H A : μ after < μ before (the mean blood pressure is less after using the drug)

If the p-value of the test is less than some significance level (e.g. α = .05), then he can reject the null hypothesis and conclude that the new drug leads to reduced blood pressure.

Example 3: Advertising Spend

Hypothesis tests are often used in business to determine whether or not some new advertising campaign, marketing technique, etc. causes increased sales.

For example, suppose a company believes that spending more money on digital advertising leads to increased sales. To test this, the company may increase money spent on digital advertising during a two-month period and collect data to see if overall sales have increased.

They may perform a hypothesis test using the following hypotheses:

• H 0 : μ after = μ before (the mean sales is the same before and after spending more on advertising)
• H A : μ after > μ before (the mean sales increased after spending more on advertising)

If the p-value of the test is less than some significance level (e.g. α = .05), then the company can reject the null hypothesis and conclude that increased digital advertising leads to increased sales.

Example 4: Manufacturing

Hypothesis tests are also used often in manufacturing plants to determine if some new process, technique, method, etc. causes a change in the number of defective products produced.

For example, suppose a certain manufacturing plant wants to test whether or not some new method changes the number of defective widgets produced per month, which is currently 250. To test this, they may measure the mean number of defective widgets produced before and after using the new method for one month.

They can then perform a hypothesis test using the following hypotheses:

• H 0 : μ after = μ before (the mean number of defective widgets is the same before and after using the new method)
• H A : μ after ≠ μ before (the mean number of defective widgets produced is different before and after using the new method)

If the p-value of the test is less than some significance level (e.g. α = .05), then the plant can reject the null hypothesis and conclude that the new method leads to a change in the number of defective widgets produced per month.

Additional Resources

Introduction to Hypothesis Testing Introduction to the One Sample t-test Introduction to the Two Sample t-test Introduction to the Paired Samples t-test

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Hey there. My name is Zach Bobbitt. I have a Masters of Science degree in Applied Statistics and I’ve worked on machine learning algorithms for professional businesses in both healthcare and retail. I’m passionate about statistics, machine learning, and data visualization and I created Statology to be a resource for both students and teachers alike.  My goal with this site is to help you learn statistics through using simple terms, plenty of real-world examples, and helpful illustrations.

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Statistics and probability

Course: statistics and probability   >   unit 12, hypothesis testing and p-values.

• One-tailed and two-tailed tests
• Z-statistics vs. T-statistics
• Small sample hypothesis test
• Large sample proportion hypothesis testing

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Video transcript

Tutorial Playlist

Statistics tutorial, everything you need to know about the probability density function in statistics, the best guide to understand central limit theorem, an in-depth guide to measures of central tendency : mean, median and mode, the ultimate guide to understand conditional probability.

A Comprehensive Look at Percentile in Statistics

The Best Guide to Understand Bayes Theorem

Everything you need to know about the normal distribution, an in-depth explanation of cumulative distribution function, a complete guide to chi-square test, what is hypothesis testing in statistics types and examples, understanding the fundamentals of arithmetic and geometric progression, the definitive guide to understand spearman’s rank correlation, mean squared error: overview, examples, concepts and more, all you need to know about the empirical rule in statistics, the complete guide to skewness and kurtosis, a holistic look at bernoulli distribution.

All You Need to Know About Bias in Statistics

A Complete Guide to Get a Grasp of Time Series Analysis

The Key Differences Between Z-Test Vs. T-Test

The Complete Guide to Understand Pearson's Correlation

A complete guide on the types of statistical studies, everything you need to know about poisson distribution, your best guide to understand correlation vs. regression, the most comprehensive guide for beginners on what is correlation, what is hypothesis testing in statistics types and examples.

Lesson 10 of 24 By Avijeet Biswal

Table of Contents

In today’s data-driven world, decisions are based on data all the time. Hypothesis plays a crucial role in that process, whether it may be making business decisions, in the health sector, academia, or in quality improvement. Without hypothesis & hypothesis tests, you risk drawing the wrong conclusions and making bad decisions. In this tutorial, you will look at Hypothesis Testing in Statistics.

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What Is Hypothesis Testing in Statistics?

Hypothesis Testing is a type of statistical analysis in which you put your assumptions about a population parameter to the test. It is used to estimate the relationship between 2 statistical variables.

Let's discuss few examples of statistical hypothesis from real-life - 

• A teacher assumes that 60% of his college's students come from lower-middle-class families.
• A doctor believes that 3D (Diet, Dose, and Discipline) is 90% effective for diabetic patients.

Now that you know about hypothesis testing, look at the two types of hypothesis testing in statistics.

Hypothesis Testing Formula

Z = ( x̅ – μ0 ) / (σ /√n)

• Here, x̅ is the sample mean,
• μ0 is the population mean,
• σ is the standard deviation,
• n is the sample size.

How Hypothesis Testing Works?

An analyst performs hypothesis testing on a statistical sample to present evidence of the plausibility of the null hypothesis. Measurements and analyses are conducted on a random sample of the population to test a theory. Analysts use a random population sample to test two hypotheses: the null and alternative hypotheses.

The null hypothesis is typically an equality hypothesis between population parameters; for example, a null hypothesis may claim that the population means return equals zero. The alternate hypothesis is essentially the inverse of the null hypothesis (e.g., the population means the return is not equal to zero). As a result, they are mutually exclusive, and only one can be correct. One of the two possibilities, however, will always be correct.

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Null Hypothesis and Alternate Hypothesis

The Null Hypothesis is the assumption that the event will not occur. A null hypothesis has no bearing on the study's outcome unless it is rejected.

H0 is the symbol for it, and it is pronounced H-naught.

The Alternate Hypothesis is the logical opposite of the null hypothesis. The acceptance of the alternative hypothesis follows the rejection of the null hypothesis. H1 is the symbol for it.

Let's understand this with an example.

A sanitizer manufacturer claims that its product kills 95 percent of germs on average. 

To put this company's claim to the test, create a null and alternate hypothesis.

H0 (Null Hypothesis): Average = 95%.

Alternative Hypothesis (H1): The average is less than 95%.

Another straightforward example to understand this concept is determining whether or not a coin is fair and balanced. The null hypothesis states that the probability of a show of heads is equal to the likelihood of a show of tails. In contrast, the alternate theory states that the probability of a show of heads and tails would be very different.

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Hypothesis Testing Calculation With Examples

Let's consider a hypothesis test for the average height of women in the United States. Suppose our null hypothesis is that the average height is 5'4". We gather a sample of 100 women and determine that their average height is 5'5". The standard deviation of population is 2.

To calculate the z-score, we would use the following formula:

z = ( x̅ – μ0 ) / (σ /√n)

z = (5'5" - 5'4") / (2" / √100)

z = 0.5 / (0.045)

We will reject the null hypothesis as the z-score of 11.11 is very large and conclude that there is evidence to suggest that the average height of women in the US is greater than 5'4".

Steps of Hypothesis Testing

Hypothesis testing is a statistical method to determine if there is enough evidence in a sample of data to infer that a certain condition is true for the entire population. Here’s a breakdown of the typical steps involved in hypothesis testing:

Formulate Hypotheses

• Null Hypothesis (H0): This hypothesis states that there is no effect or difference, and it is the hypothesis you attempt to reject with your test.
• Alternative Hypothesis (H1 or Ha): This hypothesis is what you might believe to be true or hope to prove true. It is usually considered the opposite of the null hypothesis.

Choose the Significance Level (α)

The significance level, often denoted by alpha (α), is the probability of rejecting the null hypothesis when it is true. Common choices for α are 0.05 (5%), 0.01 (1%), and 0.10 (10%).

Select the Appropriate Test

Choose a statistical test based on the type of data and the hypothesis. Common tests include t-tests, chi-square tests, ANOVA, and regression analysis. The selection depends on data type, distribution, sample size, and whether the hypothesis is one-tailed or two-tailed.

Collect Data

Gather the data that will be analyzed in the test. This data should be representative of the population to infer conclusions accurately.

Calculate the Test Statistic

Based on the collected data and the chosen test, calculate a test statistic that reflects how much the observed data deviates from the null hypothesis.

Determine the p-value

The p-value is the probability of observing test results at least as extreme as the results observed, assuming the null hypothesis is correct. It helps determine the strength of the evidence against the null hypothesis.

Make a Decision

Compare the p-value to the chosen significance level:

• If the p-value ≤ α: Reject the null hypothesis, suggesting sufficient evidence in the data supports the alternative hypothesis.
• If the p-value > α: Do not reject the null hypothesis, suggesting insufficient evidence to support the alternative hypothesis.

Report the Results

Present the findings from the hypothesis test, including the test statistic, p-value, and the conclusion about the hypotheses.

Perform Post-hoc Analysis (if necessary)

Depending on the results and the study design, further analysis may be needed to explore the data more deeply or to address multiple comparisons if several hypotheses were tested simultaneously.

Types of Hypothesis Testing

To determine whether a discovery or relationship is statistically significant, hypothesis testing uses a z-test. It usually checks to see if two means are the same (the null hypothesis). Only when the population standard deviation is known and the sample size is 30 data points or more, can a z-test be applied.

A statistical test called a t-test is employed to compare the means of two groups. To determine whether two groups differ or if a procedure or treatment affects the population of interest, it is frequently used in hypothesis testing.

Chi-Square 

You utilize a Chi-square test for hypothesis testing concerning whether your data is as predicted. To determine if the expected and observed results are well-fitted, the Chi-square test analyzes the differences between categorical variables from a random sample. The test's fundamental premise is that the observed values in your data should be compared to the predicted values that would be present if the null hypothesis were true.

Hypothesis Testing and Confidence Intervals

Both confidence intervals and hypothesis tests are inferential techniques that depend on approximating the sample distribution. Data from a sample is used to estimate a population parameter using confidence intervals. Data from a sample is used in hypothesis testing to examine a given hypothesis. We must have a postulated parameter to conduct hypothesis testing.

Bootstrap distributions and randomization distributions are created using comparable simulation techniques. The observed sample statistic is the focal point of a bootstrap distribution, whereas the null hypothesis value is the focal point of a randomization distribution.

A variety of feasible population parameter estimates are included in confidence ranges. In this lesson, we created just two-tailed confidence intervals. There is a direct connection between these two-tail confidence intervals and these two-tail hypothesis tests. The results of a two-tailed hypothesis test and two-tailed confidence intervals typically provide the same results. In other words, a hypothesis test at the 0.05 level will virtually always fail to reject the null hypothesis if the 95% confidence interval contains the predicted value. A hypothesis test at the 0.05 level will nearly certainly reject the null hypothesis if the 95% confidence interval does not include the hypothesized parameter.

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Simple and Composite Hypothesis Testing

Depending on the population distribution, you can classify the statistical hypothesis into two types.

Simple Hypothesis: A simple hypothesis specifies an exact value for the parameter.

Composite Hypothesis: A composite hypothesis specifies a range of values.

A company is claiming that their average sales for this quarter are 1000 units. This is an example of a simple hypothesis.

Suppose the company claims that the sales are in the range of 900 to 1000 units. Then this is a case of a composite hypothesis.

One-Tailed and Two-Tailed Hypothesis Testing

The One-Tailed test, also called a directional test, considers a critical region of data that would result in the null hypothesis being rejected if the test sample falls into it, inevitably meaning the acceptance of the alternate hypothesis.

In a one-tailed test, the critical distribution area is one-sided, meaning the test sample is either greater or lesser than a specific value.

In two tails, the test sample is checked to be greater or less than a range of values in a Two-Tailed test, implying that the critical distribution area is two-sided.

If the sample falls within this range, the alternate hypothesis will be accepted, and the null hypothesis will be rejected.

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Right Tailed Hypothesis Testing

If the larger than (>) sign appears in your hypothesis statement, you are using a right-tailed test, also known as an upper test. Or, to put it another way, the disparity is to the right. For instance, you can contrast the battery life before and after a change in production. Your hypothesis statements can be the following if you want to know if the battery life is longer than the original (let's say 90 hours):

• The null hypothesis is (H0 <= 90) or less change.
• A possibility is that battery life has risen (H1) > 90.

The crucial point in this situation is that the alternate hypothesis (H1), not the null hypothesis, decides whether you get a right-tailed test.

Left Tailed Hypothesis Testing

Alternative hypotheses that assert the true value of a parameter is lower than the null hypothesis are tested with a left-tailed test; they are indicated by the asterisk "<".

Suppose H0: mean = 50 and H1: mean not equal to 50

According to the H1, the mean can be greater than or less than 50. This is an example of a Two-tailed test.

In a similar manner, if H0: mean >=50, then H1: mean <50

Here the mean is less than 50. It is called a One-tailed test.

Type 1 and Type 2 Error

A hypothesis test can result in two types of errors.

Type 1 Error: A Type-I error occurs when sample results reject the null hypothesis despite being true.

Type 2 Error: A Type-II error occurs when the null hypothesis is not rejected when it is false, unlike a Type-I error.

Suppose a teacher evaluates the examination paper to decide whether a student passes or fails.

H0: Student has passed

H1: Student has failed

Type I error will be the teacher failing the student [rejects H0] although the student scored the passing marks [H0 was true]. 

Type II error will be the case where the teacher passes the student [do not reject H0] although the student did not score the passing marks [H1 is true].

Level of Significance

The alpha value is a criterion for determining whether a test statistic is statistically significant. In a statistical test, Alpha represents an acceptable probability of a Type I error. Because alpha is a probability, it can be anywhere between 0 and 1. In practice, the most commonly used alpha values are 0.01, 0.05, and 0.1, which represent a 1%, 5%, and 10% chance of a Type I error, respectively (i.e. rejecting the null hypothesis when it is in fact correct).

A p-value is a metric that expresses the likelihood that an observed difference could have occurred by chance. As the p-value decreases the statistical significance of the observed difference increases. If the p-value is too low, you reject the null hypothesis.

Here you have taken an example in which you are trying to test whether the new advertising campaign has increased the product's sales. The p-value is the likelihood that the null hypothesis, which states that there is no change in the sales due to the new advertising campaign, is true. If the p-value is .30, then there is a 30% chance that there is no increase or decrease in the product's sales.  If the p-value is 0.03, then there is a 3% probability that there is no increase or decrease in the sales value due to the new advertising campaign. As you can see, the lower the p-value, the chances of the alternate hypothesis being true increases, which means that the new advertising campaign causes an increase or decrease in sales.

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Why Is Hypothesis Testing Important in Research Methodology?

Hypothesis testing is crucial in research methodology for several reasons:

• Provides evidence-based conclusions: It allows researchers to make objective conclusions based on empirical data, providing evidence to support or refute their research hypotheses.
• Supports decision-making: It helps make informed decisions, such as accepting or rejecting a new treatment, implementing policy changes, or adopting new practices.
• Adds rigor and validity: It adds scientific rigor to research using statistical methods to analyze data, ensuring that conclusions are based on sound statistical evidence.
• Contributes to the advancement of knowledge: By testing hypotheses, researchers contribute to the growth of knowledge in their respective fields by confirming existing theories or discovering new patterns and relationships.

When Did Hypothesis Testing Begin?

Hypothesis testing as a formalized process began in the early 20th century, primarily through the work of statisticians such as Ronald A. Fisher, Jerzy Neyman, and Egon Pearson. The development of hypothesis testing is closely tied to the evolution of statistical methods during this period.

• Ronald A. Fisher (1920s): Fisher was one of the key figures in developing the foundation for modern statistical science. In the 1920s, he introduced the concept of the null hypothesis in his book "Statistical Methods for Research Workers" (1925). Fisher also developed significance testing to examine the likelihood of observing the collected data if the null hypothesis were true. He introduced p-values to determine the significance of the observed results.
• Neyman-Pearson Framework (1930s): Jerzy Neyman and Egon Pearson built on Fisher’s work and formalized the process of hypothesis testing even further. In the 1930s, they introduced the concepts of Type I and Type II errors and developed a decision-making framework widely used in hypothesis testing today. Their approach emphasized the balance between these errors and introduced the concepts of the power of a test and the alternative hypothesis.

The dialogue between Fisher's and Neyman-Pearson's approaches shaped the methods and philosophy of statistical hypothesis testing used today. Fisher emphasized the evidential interpretation of the p-value. At the same time, Neyman and Pearson advocated for a decision-theoretical approach in which hypotheses are either accepted or rejected based on pre-determined significance levels and power considerations.

The application and methodology of hypothesis testing have since become a cornerstone of statistical analysis across various scientific disciplines, marking a significant statistical development.

Limitations of Hypothesis Testing

Hypothesis testing has some limitations that researchers should be aware of:

• It cannot prove or establish the truth: Hypothesis testing provides evidence to support or reject a hypothesis, but it cannot confirm the absolute truth of the research question.
• Results are sample-specific: Hypothesis testing is based on analyzing a sample from a population, and the conclusions drawn are specific to that particular sample.
• Possible errors: During hypothesis testing, there is a chance of committing type I error (rejecting a true null hypothesis) or type II error (failing to reject a false null hypothesis).
• Assumptions and requirements: Different tests have specific assumptions and requirements that must be met to accurately interpret results.

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After reading this tutorial, you would have a much better understanding of hypothesis testing, one of the most important concepts in the field of Data Science . The majority of hypotheses are based on speculation about observed behavior, natural phenomena, or established theories.

If you are interested in statistics of data science and skills needed for such a career, you ought to explore the Post Graduate Program in Data Science.

If you have any questions regarding this ‘Hypothesis Testing In Statistics’ tutorial, do share them in the comment section. Our subject matter expert will respond to your queries. Happy learning!

1. What is hypothesis testing in statistics with example?

Hypothesis testing is a statistical method used to determine if there is enough evidence in a sample data to draw conclusions about a population. It involves formulating two competing hypotheses, the null hypothesis (H0) and the alternative hypothesis (Ha), and then collecting data to assess the evidence. An example: testing if a new drug improves patient recovery (Ha) compared to the standard treatment (H0) based on collected patient data.

2. What is H0 and H1 in statistics?

In statistics, H0​ and H1​ represent the null and alternative hypotheses. The null hypothesis, H0​, is the default assumption that no effect or difference exists between groups or conditions. The alternative hypothesis, H1​, is the competing claim suggesting an effect or a difference. Statistical tests determine whether to reject the null hypothesis in favor of the alternative hypothesis based on the data.

3. What is a simple hypothesis with an example?

A simple hypothesis is a specific statement predicting a single relationship between two variables. It posits a direct and uncomplicated outcome. For example, a simple hypothesis might state, "Increased sunlight exposure increases the growth rate of sunflowers." Here, the hypothesis suggests a direct relationship between the amount of sunlight (independent variable) and the growth rate of sunflowers (dependent variable), with no additional variables considered.

4. What are the 2 types of hypothesis testing?

• One-tailed (or one-sided) test: Tests for the significance of an effect in only one direction, either positive or negative.
• Two-tailed (or two-sided) test: Tests for the significance of an effect in both directions, allowing for the possibility of a positive or negative effect.

The choice between one-tailed and two-tailed tests depends on the specific research question and the directionality of the expected effect.

5. What are the 3 major types of hypothesis?

The three major types of hypotheses are:

• Null Hypothesis (H0): Represents the default assumption, stating that there is no significant effect or relationship in the data.
• Alternative Hypothesis (Ha): Contradicts the null hypothesis and proposes a specific effect or relationship that researchers want to investigate.
• Nondirectional Hypothesis: An alternative hypothesis that doesn't specify the direction of the effect, leaving it open for both positive and negative possibilities.

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About the Author

Avijeet is a Senior Research Analyst at Simplilearn. Passionate about Data Analytics, Machine Learning, and Deep Learning, Avijeet is also interested in politics, cricket, and football.

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• How to Write a Strong Hypothesis | Steps & Examples

How to Write a Strong Hypothesis | Steps & Examples

Published on May 6, 2022 by Shona McCombes . Revised on November 20, 2023.

A hypothesis is a statement that can be tested by scientific research. If you want to test a relationship between two or more variables, you need to write hypotheses before you start your experiment or data collection .

Example: Hypothesis

Daily apple consumption leads to fewer doctor’s visits.

Table of contents

What is a hypothesis, developing a hypothesis (with example), hypothesis examples, other interesting articles, frequently asked questions about writing hypotheses.

A hypothesis states your predictions about what your research will find. It is a tentative answer to your research question that has not yet been tested. For some research projects, you might have to write several hypotheses that address different aspects of your research question.

A hypothesis is not just a guess – it should be based on existing theories and knowledge. It also has to be testable, which means you can support or refute it through scientific research methods (such as experiments, observations and statistical analysis of data).

Variables in hypotheses

Hypotheses propose a relationship between two or more types of variables .

• An independent variable is something the researcher changes or controls.
• A dependent variable is something the researcher observes and measures.

If there are any control variables , extraneous variables , or confounding variables , be sure to jot those down as you go to minimize the chances that research bias  will affect your results.

In this example, the independent variable is exposure to the sun – the assumed cause . The dependent variable is the level of happiness – the assumed effect .

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Step 1. Ask a question

Writing a hypothesis begins with a research question that you want to answer. The question should be focused, specific, and researchable within the constraints of your project.

Step 2. Do some preliminary research

Your initial answer to the question should be based on what is already known about the topic. Look for theories and previous studies to help you form educated assumptions about what your research will find.

At this stage, you might construct a conceptual framework to ensure that you’re embarking on a relevant topic . This can also help you identify which variables you will study and what you think the relationships are between them. Sometimes, you’ll have to operationalize more complex constructs.

Step 3. Formulate your hypothesis

Now you should have some idea of what you expect to find. Write your initial answer to the question in a clear, concise sentence.

4. Refine your hypothesis

You need to make sure your hypothesis is specific and testable. There are various ways of phrasing a hypothesis, but all the terms you use should have clear definitions, and the hypothesis should contain:

• The relevant variables
• The specific group being studied
• The predicted outcome of the experiment or analysis

5. Phrase your hypothesis in three ways

To identify the variables, you can write a simple prediction in  if…then form. The first part of the sentence states the independent variable and the second part states the dependent variable.

In academic research, hypotheses are more commonly phrased in terms of correlations or effects, where you directly state the predicted relationship between variables.

If you are comparing two groups, the hypothesis can state what difference you expect to find between them.

6. Write a null hypothesis

If your research involves statistical hypothesis testing , you will also have to write a null hypothesis . The null hypothesis is the default position that there is no association between the variables. The null hypothesis is written as H 0 , while the alternative hypothesis is H 1 or H a .

• H 0 : The number of lectures attended by first-year students has no effect on their final exam scores.
• H 1 : The number of lectures attended by first-year students has a positive effect on their final exam scores.

If you want to know more about the research process , methodology , research bias , or statistics , make sure to check out some of our other articles with explanations and examples.

• Sampling methods
• Simple random sampling
• Stratified sampling
• Cluster sampling
• Likert scales
• Reproducibility

 Statistics

• Null hypothesis
• Statistical power
• Probability distribution
• Effect size
• Poisson distribution

Research bias

• Optimism bias
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• Implicit bias
• Hawthorne effect
• Anchoring bias
• Explicit bias

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A hypothesis is not just a guess — it should be based on existing theories and knowledge. It also has to be testable, which means you can support or refute it through scientific research methods (such as experiments, observations and statistical analysis of data).

Null and alternative hypotheses are used in statistical hypothesis testing . The null hypothesis of a test always predicts no effect or no relationship between variables, while the alternative hypothesis states your research prediction of an effect or relationship.

Hypothesis testing is a formal procedure for investigating our ideas about the world using statistics. It is used by scientists to test specific predictions, called hypotheses , by calculating how likely it is that a pattern or relationship between variables could have arisen by chance.

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Hypothesis testing involves formulating assumptions about population parameters based on sample statistics and rigorously evaluating these assumptions against empirical evidence. This article sheds light on the significance of hypothesis testing and the critical steps involved in the process.

What is Hypothesis Testing?

A hypothesis is an assumption or idea, specifically a statistical claim about an unknown population parameter. For example, a judge assumes a person is innocent and verifies this by reviewing evidence and hearing testimony before reaching a verdict.

Hypothesis testing is a statistical method that is used to make a statistical decision using experimental data. Hypothesis testing is basically an assumption that we make about a population parameter. It evaluates two mutually exclusive statements about a population to determine which statement is best supported by the sample data. 

To test the validity of the claim or assumption about the population parameter:

• A sample is drawn from the population and analyzed.
• The results of the analysis are used to decide whether the claim is true or not.

Example: You say an average height in the class is 30 or a boy is taller than a girl. All of these is an assumption that we are assuming, and we need some statistical way to prove these. We need some mathematical conclusion whatever we are assuming is true.

Defining Hypotheses

• Null hypothesis (H 0 ): In statistics, the null hypothesis is a general statement or default position that there is no relationship between two measured cases or no relationship among groups. In other words, it is a basic assumption or made based on the problem knowledge. Example : A company’s mean production is 50 units/per da H 0 : [Tex]\mu [/Tex] = 50.
• Alternative hypothesis (H 1 ): The alternative hypothesis is the hypothesis used in hypothesis testing that is contrary to the null hypothesis.  Example: A company’s production is not equal to 50 units/per day i.e. H 1 : [Tex]\mu [/Tex] [Tex]\ne [/Tex] 50.

Key Terms of Hypothesis Testing

• Level of significance : It refers to the degree of significance in which we accept or reject the null hypothesis. 100% accuracy is not possible for accepting a hypothesis, so we, therefore, select a level of significance that is usually 5%. This is normally denoted with  [Tex]\alpha[/Tex] and generally, it is 0.05 or 5%, which means your output should be 95% confident to give a similar kind of result in each sample.
• P-value: The P value , or calculated probability, is the probability of finding the observed/extreme results when the null hypothesis(H0) of a study-given problem is true. If your P-value is less than the chosen significance level then you reject the null hypothesis i.e. accept that your sample claims to support the alternative hypothesis.
• Test Statistic: The test statistic is a numerical value calculated from sample data during a hypothesis test, used to determine whether to reject the null hypothesis. It is compared to a critical value or p-value to make decisions about the statistical significance of the observed results.
• Critical value : The critical value in statistics is a threshold or cutoff point used to determine whether to reject the null hypothesis in a hypothesis test.
• Degrees of freedom: Degrees of freedom are associated with the variability or freedom one has in estimating a parameter. The degrees of freedom are related to the sample size and determine the shape.

Why do we use Hypothesis Testing?

Hypothesis testing is an important procedure in statistics. Hypothesis testing evaluates two mutually exclusive population statements to determine which statement is most supported by sample data. When we say that the findings are statistically significant, thanks to hypothesis testing. 

One-Tailed and Two-Tailed Test

One tailed test focuses on one direction, either greater than or less than a specified value. We use a one-tailed test when there is a clear directional expectation based on prior knowledge or theory. The critical region is located on only one side of the distribution curve. If the sample falls into this critical region, the null hypothesis is rejected in favor of the alternative hypothesis.

One-Tailed Test

There are two types of one-tailed test:

• Left-Tailed (Left-Sided) Test: The alternative hypothesis asserts that the true parameter value is less than the null hypothesis. Example: H 0 ​: [Tex]\mu \geq 50 [/Tex] and H 1 : [Tex]\mu < 50 [/Tex]
• Right-Tailed (Right-Sided) Test : The alternative hypothesis asserts that the true parameter value is greater than the null hypothesis. Example: H 0 : [Tex]\mu \leq50 [/Tex] and H 1 : [Tex]\mu > 50 [/Tex]

Two-Tailed Test

A two-tailed test considers both directions, greater than and less than a specified value.We use a two-tailed test when there is no specific directional expectation, and want to detect any significant difference.

Example: H 0 : [Tex]\mu = [/Tex] 50 and H 1 : [Tex]\mu \neq 50 [/Tex]

To delve deeper into differences into both types of test: Refer to link

What are Type 1 and Type 2 errors in Hypothesis Testing?

In hypothesis testing, Type I and Type II errors are two possible errors that researchers can make when drawing conclusions about a population based on a sample of data. These errors are associated with the decisions made regarding the null hypothesis and the alternative hypothesis.

• Type I error: When we reject the null hypothesis, although that hypothesis was true. Type I error is denoted by alpha( [Tex]\alpha [/Tex] ).
• Type II errors : When we accept the null hypothesis, but it is false. Type II errors are denoted by beta( [Tex]\beta [/Tex] ).

How does Hypothesis Testing work?

Step 1: define null and alternative hypothesis.

State the null hypothesis ( [Tex]H_0 [/Tex] ), representing no effect, and the alternative hypothesis ( [Tex]H_1 [/Tex] ​), suggesting an effect or difference.

We first identify the problem about which we want to make an assumption keeping in mind that our assumption should be contradictory to one another, assuming Normally distributed data.

Step 2 – Choose significance level

Select a significance level ( [Tex]\alpha [/Tex] ), typically 0.05, to determine the threshold for rejecting the null hypothesis. It provides validity to our hypothesis test, ensuring that we have sufficient data to back up our claims. Usually, we determine our significance level beforehand of the test. The p-value is the criterion used to calculate our significance value.

Step 3 – Collect and Analyze data.

Gather relevant data through observation or experimentation. Analyze the data using appropriate statistical methods to obtain a test statistic.

Step 4-Calculate Test Statistic

The data for the tests are evaluated in this step we look for various scores based on the characteristics of data. The choice of the test statistic depends on the type of hypothesis test being conducted.

There are various hypothesis tests, each appropriate for various goal to calculate our test. This could be a Z-test , Chi-square , T-test , and so on.

• Z-test : If population means and standard deviations are known. Z-statistic is commonly used.
• t-test : If population standard deviations are unknown. and sample size is small than t-test statistic is more appropriate.
• Chi-square test : Chi-square test is used for categorical data or for testing independence in contingency tables
• F-test : F-test is often used in analysis of variance (ANOVA) to compare variances or test the equality of means across multiple groups.

We have a smaller dataset, So, T-test is more appropriate to test our hypothesis.

T-statistic is a measure of the difference between the means of two groups relative to the variability within each group. It is calculated as the difference between the sample means divided by the standard error of the difference. It is also known as the t-value or t-score.

Step 5 – Comparing Test Statistic:

In this stage, we decide where we should accept the null hypothesis or reject the null hypothesis. There are two ways to decide where we should accept or reject the null hypothesis.

Method A: Using Crtical values

Comparing the test statistic and tabulated critical value we have,

• If Test Statistic>Critical Value: Reject the null hypothesis.
• If Test Statistic≤Critical Value: Fail to reject the null hypothesis.

Note: Critical values are predetermined threshold values that are used to make a decision in hypothesis testing. To determine critical values for hypothesis testing, we typically refer to a statistical distribution table , such as the normal distribution or t-distribution tables based on.

Method B: Using P-values

We can also come to an conclusion using the p-value,

• If the p-value is less than or equal to the significance level i.e. ( [Tex]p\leq\alpha [/Tex] ), you reject the null hypothesis. This indicates that the observed results are unlikely to have occurred by chance alone, providing evidence in favor of the alternative hypothesis.
• If the p-value is greater than the significance level i.e. ( [Tex]p\geq \alpha[/Tex] ), you fail to reject the null hypothesis. This suggests that the observed results are consistent with what would be expected under the null hypothesis.

Note : The p-value is the probability of obtaining a test statistic as extreme as, or more extreme than, the one observed in the sample, assuming the null hypothesis is true. To determine p-value for hypothesis testing, we typically refer to a statistical distribution table , such as the normal distribution or t-distribution tables based on.

Step 7- Interpret the Results

At last, we can conclude our experiment using method A or B.

Calculating test statistic

To validate our hypothesis about a population parameter we use statistical functions . We use the z-score, p-value, and level of significance(alpha) to make evidence for our hypothesis for normally distributed data .

1. Z-statistics:

When population means and standard deviations are known.

[Tex]z = \frac{\bar{x} – \mu}{\frac{\sigma}{\sqrt{n}}}[/Tex]

• [Tex]\bar{x} [/Tex] is the sample mean,
• μ represents the population mean, 
• σ is the standard deviation
• and n is the size of the sample.

2. T-Statistics

T test is used when n<30,

t-statistic calculation is given by:

[Tex]t=\frac{x̄-μ}{s/\sqrt{n}} [/Tex]

• t = t-score,
• x̄ = sample mean
• μ = population mean,
• s = standard deviation of the sample,
• n = sample size

3. Chi-Square Test

Chi-Square Test for Independence categorical Data (Non-normally distributed) using:

[Tex]\chi^2 = \sum \frac{(O_{ij} – E_{ij})^2}{E_{ij}}[/Tex]

• [Tex]O_{ij}[/Tex] is the observed frequency in cell [Tex]{ij} [/Tex]
• i,j are the rows and columns index respectively.
• [Tex]E_{ij}[/Tex] is the expected frequency in cell [Tex]{ij}[/Tex] , calculated as : [Tex]\frac{{\text{{Row total}} \times \text{{Column total}}}}{{\text{{Total observations}}}}[/Tex]

Real life Examples of Hypothesis Testing

Let’s examine hypothesis testing using two real life situations,

Case A: D oes a New Drug Affect Blood Pressure?

Imagine a pharmaceutical company has developed a new drug that they believe can effectively lower blood pressure in patients with hypertension. Before bringing the drug to market, they need to conduct a study to assess its impact on blood pressure.

• Before Treatment: 120, 122, 118, 130, 125, 128, 115, 121, 123, 119
• After Treatment: 115, 120, 112, 128, 122, 125, 110, 117, 119, 114

Step 1 : Define the Hypothesis

• Null Hypothesis : (H 0 )The new drug has no effect on blood pressure.
• Alternate Hypothesis : (H 1 )The new drug has an effect on blood pressure.

Step 2: Define the Significance level

Let’s consider the Significance level at 0.05, indicating rejection of the null hypothesis.

If the evidence suggests less than a 5% chance of observing the results due to random variation.

Step 3 : Compute the test statistic

Using paired T-test analyze the data to obtain a test statistic and a p-value.

The test statistic (e.g., T-statistic) is calculated based on the differences between blood pressure measurements before and after treatment.

t = m/(s/√n)

• m  = mean of the difference i.e X after, X before
• s  = standard deviation of the difference (d) i.e d i ​= X after, i ​− X before,
• n  = sample size,

then, m= -3.9, s= 1.8 and n= 10

we, calculate the , T-statistic = -9 based on the formula for paired t test

Step 4: Find the p-value

The calculated t-statistic is -9 and degrees of freedom df = 9, you can find the p-value using statistical software or a t-distribution table.

thus, p-value = 8.538051223166285e-06

Step 5: Result

• If the p-value is less than or equal to 0.05, the researchers reject the null hypothesis.
• If the p-value is greater than 0.05, they fail to reject the null hypothesis.

Conclusion: Since the p-value (8.538051223166285e-06) is less than the significance level (0.05), the researchers reject the null hypothesis. There is statistically significant evidence that the average blood pressure before and after treatment with the new drug is different.

Python Implementation of Case A

Let’s create hypothesis testing with python, where we are testing whether a new drug affects blood pressure. For this example, we will use a paired T-test. We’ll use the scipy.stats library for the T-test.

Scipy is a mathematical library in Python that is mostly used for mathematical equations and computations.

We will implement our first real life problem via python,

import numpy as np from scipy import stats # Data before_treatment = np . array ([ 120 , 122 , 118 , 130 , 125 , 128 , 115 , 121 , 123 , 119 ]) after_treatment = np . array ([ 115 , 120 , 112 , 128 , 122 , 125 , 110 , 117 , 119 , 114 ]) # Step 1: Null and Alternate Hypotheses # Null Hypothesis: The new drug has no effect on blood pressure. # Alternate Hypothesis: The new drug has an effect on blood pressure. null_hypothesis = "The new drug has no effect on blood pressure." alternate_hypothesis = "The new drug has an effect on blood pressure." # Step 2: Significance Level alpha = 0.05 # Step 3: Paired T-test t_statistic , p_value = stats . ttest_rel ( after_treatment , before_treatment ) # Step 4: Calculate T-statistic manually m = np . mean ( after_treatment - before_treatment ) s = np . std ( after_treatment - before_treatment , ddof = 1 ) # using ddof=1 for sample standard deviation n = len ( before_treatment ) t_statistic_manual = m / ( s / np . sqrt ( n )) # Step 5: Decision if p_value <= alpha : decision = "Reject" else : decision = "Fail to reject" # Conclusion if decision == "Reject" : conclusion = "There is statistically significant evidence that the average blood pressure before and after treatment with the new drug is different." else : conclusion = "There is insufficient evidence to claim a significant difference in average blood pressure before and after treatment with the new drug." # Display results print ( "T-statistic (from scipy):" , t_statistic ) print ( "P-value (from scipy):" , p_value ) print ( "T-statistic (calculated manually):" , t_statistic_manual ) print ( f "Decision: { decision } the null hypothesis at alpha= { alpha } ." ) print ( "Conclusion:" , conclusion )

T-statistic (from scipy): -9.0 P-value (from scipy): 8.538051223166285e-06 T-statistic (calculated manually): -9.0 Decision: Reject the null hypothesis at alpha=0.05. Conclusion: There is statistically significant evidence that the average blood pressure before and after treatment with the new drug is different.

In the above example, given the T-statistic of approximately -9 and an extremely small p-value, the results indicate a strong case to reject the null hypothesis at a significance level of 0.05. 

• The results suggest that the new drug, treatment, or intervention has a significant effect on lowering blood pressure.
• The negative T-statistic indicates that the mean blood pressure after treatment is significantly lower than the assumed population mean before treatment.

Case B : Cholesterol level in a population

Data: A sample of 25 individuals is taken, and their cholesterol levels are measured.

Cholesterol Levels (mg/dL): 205, 198, 210, 190, 215, 205, 200, 192, 198, 205, 198, 202, 208, 200, 205, 198, 205, 210, 192, 205, 198, 205, 210, 192, 205.

Populations Mean = 200

Population Standard Deviation (σ): 5 mg/dL(given for this problem)

Step 1: Define the Hypothesis

• Null Hypothesis (H 0 ): The average cholesterol level in a population is 200 mg/dL.
• Alternate Hypothesis (H 1 ): The average cholesterol level in a population is different from 200 mg/dL.

As the direction of deviation is not given , we assume a two-tailed test, and based on a normal distribution table, the critical values for a significance level of 0.05 (two-tailed) can be calculated through the z-table and are approximately -1.96 and 1.96.

The test statistic is calculated by using the z formula Z = [Tex](203.8 – 200) / (5 \div \sqrt{25}) [/Tex] ​ and we get accordingly , Z =2.039999999999992.

Step 4: Result

Since the absolute value of the test statistic (2.04) is greater than the critical value (1.96), we reject the null hypothesis. And conclude that, there is statistically significant evidence that the average cholesterol level in the population is different from 200 mg/dL

Python Implementation of Case B

import scipy.stats as stats import math import numpy as np # Given data sample_data = np . array ( [ 205 , 198 , 210 , 190 , 215 , 205 , 200 , 192 , 198 , 205 , 198 , 202 , 208 , 200 , 205 , 198 , 205 , 210 , 192 , 205 , 198 , 205 , 210 , 192 , 205 ]) population_std_dev = 5 population_mean = 200 sample_size = len ( sample_data ) # Step 1: Define the Hypotheses # Null Hypothesis (H0): The average cholesterol level in a population is 200 mg/dL. # Alternate Hypothesis (H1): The average cholesterol level in a population is different from 200 mg/dL. # Step 2: Define the Significance Level alpha = 0.05 # Two-tailed test # Critical values for a significance level of 0.05 (two-tailed) critical_value_left = stats . norm . ppf ( alpha / 2 ) critical_value_right = - critical_value_left # Step 3: Compute the test statistic sample_mean = sample_data . mean () z_score = ( sample_mean - population_mean ) / \ ( population_std_dev / math . sqrt ( sample_size )) # Step 4: Result # Check if the absolute value of the test statistic is greater than the critical values if abs ( z_score ) > max ( abs ( critical_value_left ), abs ( critical_value_right )): print ( "Reject the null hypothesis." ) print ( "There is statistically significant evidence that the average cholesterol level in the population is different from 200 mg/dL." ) else : print ( "Fail to reject the null hypothesis." ) print ( "There is not enough evidence to conclude that the average cholesterol level in the population is different from 200 mg/dL." )

Reject the null hypothesis. There is statistically significant evidence that the average cholesterol level in the population is different from 200 mg/dL.

Limitations of Hypothesis Testing

• Although a useful technique, hypothesis testing does not offer a comprehensive grasp of the topic being studied. Without fully reflecting the intricacy or whole context of the phenomena, it concentrates on certain hypotheses and statistical significance.
• The accuracy of hypothesis testing results is contingent on the quality of available data and the appropriateness of statistical methods used. Inaccurate data or poorly formulated hypotheses can lead to incorrect conclusions.
• Relying solely on hypothesis testing may cause analysts to overlook significant patterns or relationships in the data that are not captured by the specific hypotheses being tested. This limitation underscores the importance of complimenting hypothesis testing with other analytical approaches.

Hypothesis testing stands as a cornerstone in statistical analysis, enabling data scientists to navigate uncertainties and draw credible inferences from sample data. By systematically defining null and alternative hypotheses, choosing significance levels, and leveraging statistical tests, researchers can assess the validity of their assumptions. The article also elucidates the critical distinction between Type I and Type II errors, providing a comprehensive understanding of the nuanced decision-making process inherent in hypothesis testing. The real-life example of testing a new drug’s effect on blood pressure using a paired T-test showcases the practical application of these principles, underscoring the importance of statistical rigor in data-driven decision-making.

Frequently Asked Questions (FAQs)

1. what are the 3 types of hypothesis test.

There are three types of hypothesis tests: right-tailed, left-tailed, and two-tailed. Right-tailed tests assess if a parameter is greater, left-tailed if lesser. Two-tailed tests check for non-directional differences, greater or lesser.

2.What are the 4 components of hypothesis testing?

Null Hypothesis ( [Tex]H_o [/Tex] ): No effect or difference exists. Alternative Hypothesis ( [Tex]H_1 [/Tex] ): An effect or difference exists. Significance Level ( [Tex]\alpha [/Tex] ): Risk of rejecting null hypothesis when it’s true (Type I error). Test Statistic: Numerical value representing observed evidence against null hypothesis.

3.What is hypothesis testing in ML?

Statistical method to evaluate the performance and validity of machine learning models. Tests specific hypotheses about model behavior, like whether features influence predictions or if a model generalizes well to unseen data.

4.What is the difference between Pytest and hypothesis in Python?

Pytest purposes general testing framework for Python code while Hypothesis is a Property-based testing framework for Python, focusing on generating test cases based on specified properties of the code.

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Which criterion for statistical hypothesis testing can be applied in order to test for statistical significance of the regression coefficient? Fisher's F-test Gaussian Z-test Pearson's Chi-square test Student's t-test

The criterion for statistical hypothesis testing that can be applied to test for the statistical significance of a regression coefficient is Student's t-test. In regression analysis, the statistical significance of a regression coefficient is often assessed to determine if there is a significant relationship between the independent variable(s) and the dependent variable. Student's t-test is the appropriate criterion for this purpose. The t-test assesses whether the estimated regression coefficient is significantly different from zero. It compares the magnitude of the estimated coefficient to its standard error, taking into account the sample size and the variability of the data. The resulting t-statistic is then compared to the critical value from the t-distribution, based on the desired significance level and degrees of freedom. If the calculated t-statistic exceeds the critical value, it indicates that the regression coefficient is statistically significant at the chosen level of significance. This means that the coefficient is unlikely to be zero and provides evidence of a significant relationship between the variables. Therefore, Student's t-test is used to test the statistical significance of regression coefficients. Learn more about Regression coefficient click here:brainly.com/question/10869805 #SPJ11

Related Questions

1. A learning curve unit cost equation takes the form Y(u) = aub for the uth unit produced. The cost to produce the 10th unit was $95, and suppose the equation corresponds to an 88% learning curve. How many units must be produced before the unit cost is $20?

39 units must be produced before the unit cost reaches $20.

The learning curve unit cost equation Y(u) = aub represents the cost to produce the uth unit. Given that the cost to produce the 10th unit is $95 and the equation corresponds to an 88% learning curve, we need to determine the number of units that must be produced before the unit cost reaches $20.

In the learning curve unit cost equation Y(u) = aub, the variable "u" represents the unit number, "a" represents the cost of the first unit, and "b" represents the learning curve index .

To solve for the number of units required to reach a specific unit cost, we can use the following steps:

Determine the learning curve index (b):

The learning curve index can be calculated using the formula: b = log(LC) / log(2), where LC is the learning curve percentage (expressed as a decimal). In this case, the learning curve index is b = log(0.88) / log(2) ≈ -0.129.

Calculate the cost of the first unit (a):

We are given that the cost to produce the 10th unit is $95. By substituting u = 10 in the equation, we can solve for a: 95 = a * 10^(-0.129). Solving for a, we find a ≈ 219.85.

Determine the number of units for the unit cost to be $20:

We need to find the value of u when Y(u) = $20. Substituting $20 for Y(u) and solving for u in the equation 20 = 219.85 * u^(-0.129), we find u ≈ 38.78.

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Suppose Peter is a 17-year-old, full-time high school student. At present he is working 12 hours per week, without pay, on his father's pig farm and is actively seeking a better job which he would take immediately. According to the BLS household survey, Peter would be classified as Select one: A. not in the labor force. B. not in the working-age population. C. unemployed. D. employed.

According to the information provided, Peter is actively seeking a better job and is currently working 12 hours per week on his father's pig farm. Based on this, Peter would be classified as:

C. unemployed.

Being unemployed means that an individual is actively seeking employment but is currently without a job. In this case, Peter is actively looking for a better job, indicating that he is unemployed.

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When conducting an expansionary open-market purchase, the Fed government bonds, interest rates....... aggregate demand and the price level a) purchases; decreases; increases; increases b) purchases, increases; increases: increases O c) sells; decreases increases; increases d) sells; increases; decreases: decreases

The correct option is A) Purchases, decreases, increases, increases. Here is the main answer explaining why it is the correct option.The Fed is capable of conducting an expansionary monetary policy by purchasing government securities through open-market operations , which enhances the money supply in the economy.  

An open-market operation is a monetary policy that entails the buying or selling of securities on the open market by the central bank, such as the Fed, to increase or decrease the money supply in the economy. An open-market purchase is conducted by the Fed during an expansionary monetary policy, in which it purchases government bonds to increase the money supply. Through the purchase of securities , the Fed injects money into the economy, which boosts the supply of money. This increased money supply encourages banks to lend money to borrowers at a lower interest rate than they would have been able to. Borrowers borrow more because of the lower interest rates, resulting in a rise in investment and consumption . Furthermore, the expansionary policy leads to a rise in aggregate demand, as higher investment and consumption lead to an increase in demand for goods and services. Consequently, prices rise due to the increased demand.

In conclusion, conducting an expansionary open-market purchase by the Fed involves the purchase of government bonds to increase the money supply, decrease interest rates, increase consumption and investment, increase aggregate demand, and increase the price level . Therefore, option A, "Purchases, decreases, increases, increases," is the correct option.

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C2, D1, D3 Read the following case and answer the following question: Raid was born in Germany in 1985, his father gave him a building in Berlin. At the end of April, he decided to come to the UK after selling his building in Berlin. On the same date, he purchased a complex in London, and he use one of the flats at the complex as his home. Three months later he decided to go outside the UK for a 12- month trip. Question: Do you think that Raid for taxation purposes would be considered as a UK resident for the year 21/22 and why?

Raid would not be considered a UK resident for taxation purposes in the year 21/22.

Residency for taxation purposes is determined by various factors, including the number of days spent in a country, the individual's intention to stay, and their ties to that country. In this case, Raid's presence in the UK is limited to purchasing a complex and using one of the flats as his home. However, he leaves the UK three months later for a 12-month trip.

This suggests that Raid does not have a long-term intention to stay in the UK and is not establishing substantial ties to the country . Additionally, Raid's primary residence is in Berlin, as he sold his building there and only temporarily used a flat in London . Therefore, based on the information provided, Raid would not meet the criteria to be considered a UK resident for taxation purposes in the year 21/22.

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Keesha Co, borrows $260,000 cash on November 1 of the current year by signing a 120-day, 9%, $260,000 note, 1. On what date does this note mature? 2.&3. What is the amount of interest expense in the current year and the following year from this note? 4. Prepare journal entries to record(a) issuance of the note, (b) accrual of Interest on December 31, and (c) payment of the note at maturity

Interest expense is the cost a person or business incurs when borrowing money from outside sources. It is an accounting of interest payments made on outstanding debt commitments , including loans, bonds, and other debt instruments .

When a business or person takes out a loan, the lender normally collects interest as payment for extending the money. The amount paid to the lender on a regular basis as interest on the outstanding debt—typically monthly or yearly—is known as the interest expenditure .

Interest expense is recorded as an expense in the financial statements of a company, specifically in the income statement.

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Robbie is a 20 year old dependent who is a full time student. Robbie has $7,800 in income from wages and $8,000 of interest income from bonds inherited from his grandmother, What is the amount of Robbie's Computed Tax Liability for the Year 2021 before consideration of tax credits or payments? USE THE TAX RATE SCHEDULES POSTED IN THE CONTENT SECTION OF O2 (since I've found errors in some others that are publicly available)! Enter your numerical answer under the following format: 1) DO NOT USE A DOLLAR SIGN 2) DO NOT USE A COMMA SEPARATOR FOR THOUSANDS IF APPLICABLE: 3) ROUND UP ANY FRACTIONAL AMOUNTS, IF APPLICABLE, TO THE NEAREST DOLLAR Answer: 765

To get the amount of Robbie's Computed Tax Liability for the Year 2021 before consideration of tax credits or payments, we need to round up any fractional amounts , if applicable, to the nearest dollar. So the answer is $765.

Robbie is a 20-year-old dependent who is a full-time student . Robbie has $7,800 in income from wages and $8,000 of interest income from bonds inherited from his grandmother. Let's calculate the Computed Tax Liability for the Year 2021 before consideration of tax credits or payments using the tax rate schedules posted in the content section of O2.Solution:Taxable Income = (Wages) + (Interest Income) = $7,800 + $8,000 = $15,800Note: Interest income is taxable, so the total income is $15,800.Tax Liability from Tax Tables2021 Tax Tables show that $9,950 is the taxable income threshold for single taxpayers. Robbie's taxable income exceeds this threshold by $5,850 ($15,800 - $9,950). Robbie will be taxed at a rate of 12% on this amount. Thus,Tax Liability = (Rate × (Taxable Income - Threshold)) + BaseTax= (12% × ($15,800 - $9,950)) + $995= $762.60 + $995= $1,757.60.To get the amount of Robbie's Computed Tax Liability for the Year 2021 before consideration of tax credits or payments, we need to round up any fractional amounts, if applicable, to the nearest dollar.So, the answer is $765. Answer: 765

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The food services division of Cedar River Amusement Park, Inc. is studying the amount families who visit the amusement park spend per day on food and drink. A sample of 40 families who visited the park yesterday revealed they spent the following amounts. $77 $19 $63 $84 $38 $54 $50 $60 $54 $56 $36 $26 $50 $34 $44 $41 $58 $58 $53 $51 $63 $62 $61 $61 $66 $ 52 $60 $61 $47 $63 $71 $66 $ 71 $63 a. How many classes will sample have? [1 mark] b. How many intervals will it have? [1 mark] c. what will be the upper and lower limit? [1 mark] d. construct a histogram and a pie chart to present the data according to classes. [7 marks]

A. The sample has 6 classes.

B. The sample has 6 intervals.

C. Lower class limits: $19, $30, $41, $52, $63, $74

Upper class limits: $30, $41, $52, $63, $74, $85

a. The number of classes in the sample can be determined using the formula:

k = 1 + 3.322 log n, where n is the sample size and k is the number of classes.

Substituting n = 40 into the formula:

k = 1 + 3.322 log 40 ≈ 1 + 3.322 × 1.602 ≈ 1 + 5.326 ≈ 6

Therefore, there are 6 classes in the sample.

b. The range of a set of values is calculated by taking the difference between the highest and lowest values in the set.

Range = Maximum value - Minimum value

= $84 - $19 = $65

Since there are six classes in the sample, dividing the range by the number of classes yields the class interval:

Class interval = Range /Number of classes

= $65/6 ≈ $11

Thus, there are six intervals in the sample.

c. The limits of each class interval can be calculated by starting with the lowest value of the set and adding the class interval repeatedly until all class intervals have been covered.

Lower class limits: $19, $30, $41, $52, $63, $74

Upper class limits : $30, $41, $52, $63, $74, $85

d. A histogram and a pie chart that represent the data in terms of classes are shown below. The y-axis of the histogram shows the frequency (number of families) in each class interval, while the x-axis shows the class intervals. The pie chart shows the proportion of families in each class interval.

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A. Equipment With A Book Value Of $82,500 And An Original Cost Of $163,000 Was Sold At A Loss Of $32,000. B. Paid $109,000 Cash For A New Truck. C. Sold Land Costing $315,000 For $415,000 Cash, Yielding A Gain Of $100,000. D. Stock Investments Were Sold For $94,900 Cash, Yielding A Gain Of $15,500. Use The Above Information To Determine Cash Flows From investing activities.

The cash flows from investing activities are as follows:

A. Cash outflow of $32,000.

B. Cash outflow of $109,000.

C. Cash inflow of $415,000.

D. Cash inflow of $94,900.

Cash refers to physical currency and coins, as well as funds held in bank accounts or other highly liquid assets that are readily accessible for transactions. It is a crucial component of a company's or individual's financial position and serves as a medium of exchange for goods, services, and debt settlement. Cash represents liquid assets that can be used to meet immediate financial obligations or invest in opportunities. Managing cash effectively is essential for maintaining liquidity, covering expenses, meeting financial obligations, and funding business operations. Cash flows, both inflows and outflows, are recorded in cash flow statements to track the movement of cash and provide insights into a company's financial health and cash management .

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Cost of​ equity: SML. Stan is expanding his business and will sell common stock for the needed funds. If the current​ risk-free rate is 4.6​% and the expected market return is 14.2​%, Question

According to the information we can infer that the correct option is 10.56% if the beta of the stock is 1.01 (option C):

To calculate the cost of equity can be determined using the Capital Asset Pricing Model (CAPM) with the following formula:

Now, we have to prove each option to identify the correct one:

For option (a):

For option (b):

For option (c):

For option (d):

According to the above we can conclude that the correct answer is option (c) 10.56% if the beta of the stock is 1.01.

Note: This question is incomplete. Here is the complete information:

Stan is expanding his business and will sell common stock for the needed funds. If the current risk-free rate is 4.3% and the expected market return is 10.5%, the cost of equity for Stan is a. 9.01% if the beta of the stock is 0.76. b. 9.94% if the beta of the stock is 0.91 c. 10.56% if the beta of the stock is 1.01. d. 1 1.37% if the beta of the stock is 1.14

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1. Skippy has the following utility function: u = xy and faces the budget constraint: M = Pxx+Pyy. (a) Use the MRS = Px/py and the budget constraint OR Lagrange method to find Skippy's demand functions, indirect utility and expenditure function. (b) Suppose M = 120, Py = 1 and P = 4. What is Skippy's optimal x, y and utility number? If the price of x was lowered to 2 what would be her x,y and utility number (c) What is the most Skippy would pay to have Pr lowered to 2? (d) Suppose M = 120, Py = 1 and Pr = 4. How much additional income would Skippy need to be as well off as if the price of a had fallen to 2?

Skippy has the following utility function: u = xy and faces the budget constraint: M = Pxx+ Pyy. The MRS is equal to Px/Py. The utility maximization problem is subject to the budget constraint. The LaGrange function is expressed as: L = xy + λ (M - Pxx - Pyy)Differentiating L with respect to x and equating to zero yields;

∂L/∂x = y - 2λPx = 0.

Differentiating L with respect to y and equating to zero yields;

Differentiating L with respect to λ and equating to zero yields;

M - Pxx - Pyy = 0

y = 2λPx and

Substituting the expression of y into the budget constraint yields;

M = Px (2λPx) + Pyy.

Substituting the expression of x into the budget constraint yields;

M = Py (2λPy) + Pxx.

Rearranging the equation;

λ = √(M/2(Px)(Py))

Substituting the value of λ into the expressions of x and y gives;

x = (M/2Px), and y = (M/2Py).

The demand functions are x = M/2Px, and y = M/2Py.The indirect utility function is equal to the maximum utility level attained by a consumer at a given price and income. The indirect utility function is given by:

V (P, M) = u (x*, y*)

= (M/2Px) (M/2Py)

= M2/(4PXPY)

The expenditure function is given by:

e (P, U) = min {Pxx + Pyy: u (x, y) = U}

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Consider a simple macro model with demand-determined output. Other things being equal, the price level and desired aggregate expenditure are related to each other O a. proportionally. O b. exponentially. O c. progressively. O d. negatively. O e. positively.

In a simple macro model with demand-determined output, the price level and desired aggregate expenditure are typically related to each other in an inversely proportional manner.

This means that as the price level increases, the desired aggregate expenditure decreases, and vice versa. This inverse relationship is known as the negative relationship between the price level and aggregate expenditure .

The reason for this relationship can be understood through the components of aggregate expenditure. Aggregate expenditure consists of consumption expenditure (C), investment expenditure (I), government expenditure (G), and net exports (NX). The consumption expenditure component is typically the largest part of aggregate expenditure and is influenced by the price level.

As the price level increases, the purchasing power of households decreases. This means that for a given level of income , consumers can afford to purchase fewer goods and services. As a result, consumption expenditure decreases, leading to a decrease in the overall desired aggregate expenditure.

Similarly, the price level affects other components of aggregate expenditure. Higher prices can increase the cost of investment projects, discourage government spending, and affect the competitiveness of exports, leading to a decrease in investment, government expenditure, and net exports, respectively.

Overall, the negative relationship between the price level and desired aggregate expenditure is a fundamental concept in macroeconomics . It illustrates how changes in the price level can impact the overall level of spending in an economy. This relationship forms the basis for analyzing the effects of monetary policy, as central banks adjust interest rates to influence the price level and stimulate or dampen aggregate expenditure to achieve macroeconomic stability.

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When a company declares and distributes stock dividends, it should be reported on financing activities section of the statement of cash flows. O True O False

The given statement is false. When a company declares a stock dividend , it does not involve any cash flow. Instead, the company issues new shares of stock to its existing shareholders in proportion to their existing holdings.

This increases the number of shares outstanding, but does not affect the company's cash balance.As a result, stock dividends are not reported on the statement of cash flows . The statement of cash flows only reports cash inflows and outflows, and stock dividends do not involve any cash.Stock dividends are reported on the statement of stockholders' equity. This statement shows the changes in a company's equity accounts over time, including changes due to stock dividends, stock splits, and other equity transactions.

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there are three objectives to this assignment: 1. to familiarize yourself with a real scheduler. 2. to change that scheduler to a new algorithm. 3. to make a graph

The objectives of the assignment are to familiarize yourself with a real scheduler, change the scheduler to a new algorithm, and create a graph.

The assignment has three objectives. Firstly, it aims to familiarize you with a real scheduler, allowing you to understand its functionalities and operation. This involves studying the existing scheduler and gaining knowledge about its algorithm and features .

Secondly, the assignment requires you to modify the scheduler by implementing a new algorithm. This task involves analyzing the current scheduler's limitations or requirements and designing and implementing a new algorithm that addresses those concerns or improves its performance in some way.

The modification should involve changing the logic or functionality of the scheduler to incorporate the new algorithm.

Lastly, the assignment requires you to create a graph. This graph could represent various aspects related to the scheduler , such as performance metrics, system utilization, or the impact of the new algorithm.

The graph will visually represent data and provide a clear understanding of the scheduler's behavior or the effects of the algorithm change.

Overall, the assignment aims to provide a practical learning experience by working with a real scheduler, applying algorithmic modifications, and visually presenting the results through a graph.

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Willow Brook National Bank operates a drive-up teller window that allows customers to complete bank transactions without getting out of their cars. On weekday mornings, arrivals to the drive-up teller window occur at random, with an arrival rate of 18 customers per hour or 0.3 customers per minute. What is the mean or expected number of customers that will arrive in a ten-minute period?

Willow Brook National Bank operates a drive-up teller window that allows customers to complete bank transactions without getting out of their cars.

The Poisson distribution is applied when events occur randomly in a given period, and the average number of events per interval is known or estimated.The expected number of customers that will arrive in a ten-minute period is 3 customers.

The calculated value of the expected number of customers can be used to estimate the probability of the number of customers who will arrive at the drive-up teller window in the future. Suppose the bank wants to find the probability of having four customers arrive at the drive-up teller window in a ten-minute period. The bank can use the Poisson distribution to estimate the probability.

The probability of having four customers arrive at the drive-up teller window in a ten-minute period can be calculated as follows: P(X = 4) = e−3(3^4 / 4!) = 0.168The bank can conclude that there is a 16.8% chance of having four customers arrive at the drive-up teller window in a ten-minute period.

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An open-ended fund has stocks of three companies: 1,406 shares of K.Kreme currently valued at $14.00. 947 shares of Ben & Jerry's currently values at $44.00 and 2,180 shares of Coke currently valued at $53.00. The fund has 3500 shares outstanding. What is the net asset value (NAV) of the fund? Enter your answer rounded to 2 decimals, and without any units. So, for example, if your answer is 64.4568 , then just enter 64.46.

The net asset value (NAV) of the fund is $ 50.54 .

To calculate the net asset value (NAV) of the fund, we need to determine the total value of the stocks held by the fund and divide it by the total number of shares outstanding.

Let's calculate the total value of each stock:

K.Kreme: 1,406 shares * $14.00/share = $19,684.00

Ben & Jerry's: 947 shares * $44.00/share = $41,668.00

Coke: 2,180 shares * $53.00/share = $115,540.00

Now, we sum up the total value of all stocks:

$19,684.00 + $41,668.00 + $115,540.00 = $ 176,892.00

Finally, we divide the total value by the number of shares outstanding to get the net asset value (NAV) of the fund:

$176,892.00 / 3,500 shares = $50.54

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To obtain a party's nomination for a seat in Congress, a candidate must typically: a. win a plurality of votes in a primary election. b. win a majority of votes in a primary election. c. obtain the party's approval after winning a primary.

To obtain a party's nomination for a seat in Congress, a candidate must typically win a plurality of votes in a primary election. primary election is a preliminary election in which voters choose their party's nominees for public office.

This is the main answer.A

The primary goal of a primary election is to narrow the field of candidates for an office or position . The winner of the primary election is usually the nominee of their respective political party. That is to say, to become a political party's nominee for Congress, a candidate must win the primary election of that party.The option a, "win a plurality of votes in a primary election" is the correct answer.

Candidates must win a plurality of votes in the primary election to obtain a party's nomination for a seat in Congress. A plurality refers to a greater number of votes than any other candidate but not necessarily a majority. In other words, to obtain a party's nomination for a seat in Congress, a candidate must win more votes than any other candidate in the primary election.

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Firm commitment versus best efferts. Astro Investment Bank offers Lunar Vacations the following options on its initial public sale of equity: (a) a best efforts arrangement whereby Astro will keep 3.1% of the retail sales or (b) a firm commitment arrangement of $10,200,000. Lunar plans on offering shares at $11.17 per share to the public. What is the break-even point in number of shares for Lunar Vacations? What are the proceeds to Lunar Vacations and Astro Investment Bank at the break-even point?

The break - even point in number of shares for Lunar Vacations is determined by dividing the total amount needed to be raised by the offer price per share.

To determine the break-even point in number of shares for Lunar Vacations, we need to divide the total amount needed to be raised by the offer price per share. In this case, the total amount needed to be raised is $ 10,200,000 and the offer price per share is $11.17. Dividing these values, we find that the break-even point is approximately 914,438 shares .

At the break-even point, the proceeds to Lunar Vacations can be calculated by multiplying the offer price per share by the number of shares sold , which is 914,438 in this case. The proceeds to Lunar Vacations would be approximately $10,220,000.

For Astro Investment Bank, in the best efforts arrangement, they keep 3.1% of the retail sales . Since the break-even point represents the point where no profit or loss is incurred, Astro Investment Bank would not receive any proceeds at the break-even point.

Therefore, the break-even point for Lunar Vacations is approximately 914,438 shares, and at this point , Lunar Vacations would receive proceeds of approximately $10,220,000, while Astro Investment Bank would not receive any proceeds in the best efforts arrangement.

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Discussion Questions 1. Why do you think cyber crime is a threatening fraud? 2. What are the advantages and disadvantages of keeping a credit/ visa or debit card? 3. Having a cash less society is one of the aims of any developing or developed country. What could be the advantages and disadvantages of carrying your card around to undertake daily transactions? 4. Why are banks putting up measures such as limited transactions in a day by a customer, even though the money is owned by the customer and he or she should be allowed to undertake any number of transactions they like? 5. When a customer has suffered losses via credit card fraud, is it only the customer's loss or the bank's loss as well? Explain why or why not? 6. Find out the maximum amount of withdrawal you can make in a day using your access card from a Fijian bank. Why is there a limit this? 7. Some emails can be sent cloning as your employer asking for your information. What should you do in such a case? 8. What is the measure you should undertake if you identify any unusual transaction from your account? Why should you undertake that measure? 9. What measures has Fiji as a country undertaken to ensure it is safe from cyber fraud? 10. After studying the lecture on cyber fraud, what measures should you undertake to ensure that you and your employer is safe from cyber fraud?

1. Cybercrime is a threatening fraud because it can result in significant financial losses , identity theft, and damage to individuals, businesses, and even governments. Cybercriminals can gain unauthorized access to sensitive information, such as personal and financial data, and use it for fraudulent activities or sell it on the dark web. They can also launch sophisticated attacks, such as ransomware or phishing, to exploit vulnerabilities in computer systems and networks. The rapid advancement of technology and increasing connectivity have made individuals and organizations more vulnerable to cyber threats, making cybercrime a serious concern.

2. Advantages and disadvantages of keeping a credit/visa or debit card:

- Advantages: Convenience and ease of use, widely accepted for payment, can help build credit history (in the case of credit cards), offers rewards and benefits (such as cashback or airline miles), can provide purchase protection and dispute resolution services.

- Disadvantages: Risk of overspending and accumulating debt (in the case of credit cards), potential for high-interest rates and fees, vulnerability to fraud and identity theft, reliance on electronic systems (which can be subject to outages or technical issues), potential for impulse buying and lack of financial discipline.

3. Advantages and disadvantages of carrying your card for daily transactions in a cashless society :

- Advantages: Convenience and efficiency in making transactions, reduced need to carry cash, ability to track and manage expenses digitally, potential for faster and more secure transactions , accessibility to online shopping and global transactions.

- Disadvantages: Dependency on technology and electronic systems, vulnerability to cyber fraud and hacking, potential for unauthorized access to personal and financial information, risk of losing the card or having it stolen, reliance on stable internet connectivity, exclusion of individuals without access to digital payment methods.

4. Banks implement measures such as limited transactions in a day by a customer to enhance security and prevent fraud. These measures are put in place to protect both the customer and the bank. By limiting the number of transactions , banks can detect and prevent suspicious activities more effectively, reducing the risk of unauthorized access and fraudulent transactions. Additionally, these measures help banks manage their operational and financial risks and maintain the overall stability of the banking system.

5. When a customer suffers losses via credit card fraud, both the customer and the bank can experience losses, although the extent may vary. Typically, banks have systems in place to detect and prevent fraudulent transactions, and they often provide reimbursement to customers for unauthorized charges. However, some banks may have specific terms and conditions regarding liability for fraudulent transactions, and customers are usually required to report the fraud promptly to minimize their liability. It is important for customers to review their bank's policies and take appropriate measures to protect their accounts and personal information .

6. The maximum amount of withdrawal a person can make in a day using an access card from a Fijian bank may vary depending on the bank and the type of account held by the individual. Limits on daily withdrawals are often implemented as a security measure to protect customers from potential fraud or theft. By setting withdrawal limits, banks can mitigate the risk of unauthorized access to funds and minimize the potential financial impact in case of card loss or theft. Additionally, withdrawal limits help banks manage liquidity and maintain efficient operations.

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The form of business that does not carry unlimited personal liability for the owner is A. A general partnership B. A sole proprietorship C. A corporation D. None. All forms of busi

Option (c), The form of business that does not carry unlimited personal liability for the owner is a corporation .

This form of business structure provides limited liability protection to the owners, meaning that their personal assets are not at risk if the business is sued or goes bankrupt.

The type of business that does not carry unlimited personal liability for the owner is a corporation. The option is C.

The word "corporation" refers to a kind of business that has a separate legal entity from its owners. A corporation is a business entity that is separate from its owners, and it can sue or be sued in its name. Corporations' owners, known as shareholders , are not personally liable for the corporation's debts or legal obligations. This is a big advantage for business owners because it means that their personal assets are not at risk if the business goes bankrupt or is sued.

A general partnership is a business structure where two or more people work together to run a business, and they share the profits and losses. In a general partnership, each partner is personally liable for the business's debts and obligations. A sole proprietorship is a business owned by one person who is responsible for all aspects of the business's operation. The owner of a sole proprietorship has unlimited personal liability for the business's debts and legal obligations. Therefore, options A and B are incorrect.

None of the above options is the correct answer to this question except option C, a corporation.

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Publisher Co. delivers 1,000 books to Bookstore Company under a consignment arrangement. The cost per book is P 300. Publisher Company pays freight of P 22 per book. Bookstore Company is entitled to a 20% commission based on the Publisher's suggested retail price. However, Bookstore Company marks up the Publisher's suggested retail price for another 15%. Six months after the end of the semester, Bookstore Company remits P 254,700 to the Publisher for the sale of 700 books, after deduction of P 69,300 for the following: 2% withholding tax based on the publisher's suggested retail price. • Bookstore's commission. Required: 1. How much profit is recognized by the Publisher? 2. How much profit is recognized by the Bookstore? 3. How much is the costs of the unsold books?

Publisher Co. delivers 1,000 books to Bookstore Company under a consignment expression arrangement. The cost per book is P 300. Publisher Company pays freight of P 22 per book. Bookstore Company is entitled to a 20% commission based on the Publisher's suggested retail price.

Bookstore Company remits P 254,700 to the Publisher for the sale of 700 books, after deduction of P 69,300 for the following: 2% withholding tax based on the publisher's suggested retail price.

Bookstore's commission.The profit is calculated as follows:1. Profit recognized by the Publisher

= Total revenue

= Total revenue - Total costs

= [(Number of sold books) × (Selling price)] - [(Number of sold books) × ( Publisher's suggested retail price)]

= [700 × (Price of the Bookstore + (0.15 × Publisher's suggested retail price))] - [700 × (Publisher's suggested retail price)]

= [700 × (1.15 × 900)] - [700 × 900] = P 598,600 - P 630,000

= - P 31,400 (loss)3. The cost of the unsold books = (Number of unsold books) × (Cost per book + Freight per book)

= [300 × (1,000 - 700)] + [22 × (1,000 - 700)] = 90,000 + 6,600

= P 96,600Therefore,1. Profit recognized by the Publisher

= P 373,2002. Profit recognized by the Bookstore

= P 31,400 (loss)3. The cost of the unsold books

= P 96,600.

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*please show all work and formula* 41. You are considering an investment with the following cash flows. If the required rate of return for this investment is 13.5%, should you accept it based solely on the internal rate of return rule?

The IRR (16.17%) to the required rate of return (13.5%), we can see that the IRR is greater than the required rate of return. Based solely on the internal rate of return rule, you should accept the investment .

To determine whether you should accept the investment based solely on the internal rate of return (IRR) rule, you need to calculate the IRR and compare it to the required rate of return.

The IRR is the discount rate at which the net present value (NPV) of the cash flows is equal to zero. If the IRR is greater than or equal to the required rate of return, then you should accept the investment.

Let's calculate the NPV of the cash flows using the given information:

Year 0: -$10,000 (initial investment)

Year 1: $4,000

Year 2: $4,500

Year 3: $5,500

Year 4: $6,000

Using the formula for NPV:

NPV = CF₀ / (1 + IRR)⁰ + CF₁ / (1 + IRR)¹ + CF₂ / (1 + IRR)² + CF₃ / (1 + IRR)³ + CF₄ / (1 + IRR)⁴.  Setting NPV to zero, we can solve for the IRR:

0 = -10,000 / (1 + IRR)⁰ + 4,000 / (1 + IRR)¹ + 4,500 / (1 + IRR)² + 5,500 / (1 + IRR)³ + 6,000 / (1 + IRR)⁴

To find the IRR, we can use trial and error , or utilize numerical methods such as Excel's IRR function. Assuming you are using a financial calculator, you can input the cash flows and solve for the IRR.

IRR ≈ 16.17%

Comparing the IRR (16.17%) to the required rate of return (13.5%), we can see that the IRR is greater than the required rate of return. Therefore, based solely on the internal rate of return rule, you should accept the investment.

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Question 36(Multiple Choice ) (04.06 MC) Assume that an economy is going through a slump and is experiencing less than ideal output levels and a decreased national income. Which one of the following actions can a central bank take in order to fix the economy? The central bank advises the government to increase taxes. The central bank increases the discount rate for commercial banks. O The central bank increases the reserve ratio of commercial banks. The central bank of the country buys securities via open market operations. The central bank of the country sells securities via open market operations. Question 34(Multiple Choice ) (04.07 MC) If an economy is experiencing equilibrium in the loanable funds market with an 8% interest rate, what are the consequences if the interest rate falls to 6%? O At lower interest rates, households will be willing to save more, and firms will be willing to invest more. O At lower interest rates, households will be willing to save more, and firms will be willing to invest less. O At lower interest rates, households will be willing to spend less, and firms will not be willing to invest more. O At lower interest rates, households will be willing to spend less, and firms will be willing to invest more. O At lower interest rates, households will be willing to spend more, and firms will be willing to invest more.

Increase in spending and investments will lead to an increase in output levels and national income , which is an indication of an improved economy.

In order to fix the economy, the central bank of the country buys securities via open market operations. This action by the central bank of the country will improve the economy of the country. .As the country's economy is experiencing less than ideal output levels and a decreased national income, the central bank needs to take some action to fix the economy. In this situation, the central bank can buy securities via open market operations. By buying securities, the central bank will increase the money supply in the country.

With more money in the market, there will be an increase in consumer spending and investments by firms. This increase in spending and investments will lead to an increase in output levels and an increase in the national income of the country .As a result, the country's economy will improve, which is the ultimate goal of the central bank . So, buying securities via open market operations is an appropriate action by the central bank to improve the economy.

Question 34:At lower interest rates, households will be willing to spend more, and firms will be willing to invest more. This statement is true when an economy is experiencing equilibrium in the loanable funds market with an 8% interest rate. .In an economy that is experiencing equilibrium in the loanable funds market with an 8% interest rate, if the interest rate falls to 6%, households will be willing to spend more, and firms will be willing to invest more.

This is because when the interest rate falls, borrowing money becomes cheaper for households and firms, leading to an increase in spending and investments. The low interest rate will also lead to an increase in the money supply in the economy. The increase in money supply will also lead to an increase in spending and investments as there will be more money available in the economy.

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who is responsible for reporting employee health issues to the appropriate authorities

The entity that is responsible for reporting employee health issues to the appropriate authorities is the employer . When it comes to maintaining and promoting the health and safety of employees, employers are legally obligated to take a proactive approach, and reporting health problems is one of the ways they can do so.

These health concerns can range from exposure to hazardous materials in the workplace to the spread of infectious diseases. If an employee experiences a health issue that is directly linked to their work, the employer must report the illness or injury to the appropriate authorities, such as the Occupational Safety and Health Administration (OSHA) or the Centers for Disease Control and Prevention (CDC).

This process is important for two reasons: first, it ensures that public health officials are aware of any potential outbreaks or risks, allowing them to take appropriate action to protect the broader population. Second, it ensures that the affected employee receives any necessary health care and compensation, allowing them to recover from the illness or injury without undue financial stress.

It is important for employers and healthcare professionals to be familiar with the specific reporting requirements applicable to their jurisdiction to ensure compliance with the relevant laws and regulations.

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Knowledge Check 01 Which of the following statements about why companies frequently choose to lease assets are true? (Select all that apply.) Check All That Apply Leasing offers flexibility when disposing of the asset. Leasing offers tax savings over outright purchases. Leasing provides protection against the risk of declining asset values. Leasing reduces the upfront cash needed to use an asset.

Only the statements 1) leasing offers flexibility when disposing of the asset, and 4) leasing reduces the upfront cash needed to use an asset are true.

the following statements about why companies frequently choose to lease assets are true:

1) leasing offers flexibility when disposing of the asset.2) leasing reduces the upfront cash needed to use an asset.

these two statements accurately reflect reasons why companies often opt for leasing. leasing provides flexibility in terms of returning or upgrading the asset at the end of the lease term. it also allows companies to use the asset without requiring a large upfront cash outlay, making it an attractive  for preserving cash flow.

the other two statements are not universally true and may vary depending on the specific circumstances:

- leasing does not necessarily offer tax savings over outright purchases . tax implications can differ based on various factors such as local tax laws, the nature of the asset, and the company's financial situation.- leasing may not always provide protection against the risk of declining asset values. the company may still bear the risk if the lease terms do not include provisions for depreciation or fluctuations in asset value.

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Corporate parenting refers to all of the following EXCEPT: Question 14 options: a. efforts to judiciously segregate funds for each business in such a way that keeps the money safe and discourages shifting funds across business units. b. efforts to capitalize on the umbrella brands and enhance value proposition across businesses. c. the corporation's ability to provide generalized support resources so as to create value by lowering companywide overhead costs by eliminating duplication of efforts. d. the help subsidiaries receive in performing better when they utilize astute high-level guidance from corporate executives. e. the role that a diversified corporation plays in nurturing its component businesses through the provision of top management expertise, disciplined control, financial resources, and capabilities.

Corporate parenting refers to all of the following EXCEPT efforts to judiciously segregate funds for each business in such a way that keeps the money safe and discourages shifting funds across business units. The correct option is a.

Corporate parenting is the act of aiding a subsidiary with the aim of adding value to the subsidiary and then generating value for the parent corporation. Corporate parenting entails the role that a diversified corporation plays in nurturing its component businesses through the provision of top management expertise, disciplined control, financial resources, and capabilities.

 a. Efforts to judiciously segregate funds for each business in such a way that keeps the money safe and discourages shifting funds across business units: This option is not a part of corporate parenting. It focuses on financial segregation and discouraging fund transfers between businesses, which is not directly related to the nurturing and support provided by a diversified corporation.

b. Efforts to capitalize on the umbrella brands and enhance value proposition across businesses: This is a part of corporate parenting. It involves leveraging the shared brand and reputation of the corporation to enhance the value proposition and market position of its subsidiary businesses.

c. The corporation's ability to provide generalized support resources to create value by lowering companywide overhead costs by eliminating duplication of efforts: This is a part of corporate parenting. It refers to the corporation's ability to provide shared resources and support services that help lower costs and eliminate duplication of efforts across its subsidiary businesses.

d. The help subsidiaries receive in performing better when they utilize astute high-level guidance from corporate executives: This is a part of corporate parenting. It highlights the role of corporate executives in providing guidance and support to the subsidiaries, enabling them to perform better.

e. The role that a diversified corporation plays in nurturing its component businesses through the provision of top management expertise, disciplined control, financial resources, and capabilities: This is a part of corporate parenting. It encompasses the overall support and nurturing provided by the diversified corporation to its subsidiary businesses, including top management expertise, control, financial resources, and capabilities.

In summary, option a is the one that does not align with the concept of corporate parenting , as it focuses on fund segregation rather than nurturing and support. Therefore option a is correct statement.

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Kelly Industries issued 7% bonds, dated January 1, with a face value of $150,000 on January 1, 2021. The bonds mature in 2031 /10 years) Interest is paid semiannually on June 30 and December 31. For bonds of similar risk and maturity the market yield is 9%. What was the issue price of the bonds? FV of $1. PV of $1. FVA of $1. PVA of $1. EVAD of $1 and PVAD of 51) (Use appropriate factor(s) from the tables provided.) Saved Helps Multiple Choice Ο $68,292. Ο $144,545. Ο $113,126. Ο $130,488.

Kelly Industries issued 7% bonds, dated January 1, with a face value of $150,000 on January 1, 2021. The bonds mature in 2031 /10 years) Interest is paid semiannually on June 30 and December 31. For bonds of similar risk and maturity, the market yield is 9%.

We have to find the issue price of the bonds.FV of $1 for n = 20 periods and i = 4.5% is 2.653 PV of $1 for n = 20 periods and i = 4.5% is 0.377 FVA of $1 for n = 20 periods and i = 4.5% is 57.275 PVA of $1 for n = 20 periods and i = 4.5% is 11.814 EVAD of $1 for n = 20 periods and i = 4.5% is 2.889 PVAD of $1 for n = 20 periods and i = 4.5% is 20.170Step 1:Calculate the periodic interest rate or semi-annual yield.r = 9/2 = 4.5%Step 2:Calculate the total number of periods or n.n = 10 years x 2 semi-annual periods per year = 20 periodsStep 3:Calculate the future value of $1 at 4.5% for 20 periods. FV = 1 x 2.653 = 2.653Step 4:Calculate the present value of $1 at 4.5% for 20 periods. PV = 1 x 0.377 = 0.377Step 5:Calculate the future value of an annuity of $1 at 4.5% for 20 periods. FVA = 1 x 57.275 = 57.275Step 6:Calculate the present value of an annuity of $1 at 4.5% for 20 periods. PVA = 1 x 11.814 = 11.814Step 7:Calculate the present value of the bond.Using the PV of an annuity of $1 table and a 9% interest rate, the present value factor for 20 periods is 9.231.The present value of the bond is:Present value of interest payments + Present value of principalPV = [57.275 x (1 - 1/1.09231)] / 0.09 + [150,000 x 0.377] / 1.09231PV = $45,919.15 + $54,569.13PV = $100,488.28Therefore, the issue price of the bonds was $100,488.28. Hence, the correct option is O $100,488.

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Joy Bionic’s Ltd. (Bionic’s) among other things sells robot kits and robotic systems. One of the replacement items that it holds in inventory is a "robocontroller". Bionic’s sells 1,125 robocontrollers annually. The cost to place an order for robocontrollers is $4.00, while the holding cost is 14.4% of the selling price. Joy purchases and takes delivery of the robocontroller from a supplier for an all in cost of $55. Joy operates 7 days a week working 50 weeks per year. The selling price of the robocontroller is $100. If a sale cannot be made because there are no controllers on hand, Joy forgoes the contribution margin on the robocontroller. Lead time on orders is 1 week. Bionic’s sales have been summarized and statistically analyzed the sales profile is as follows: Sales (units) 5 10 15 20 25 30 34 Demand Probalility 0.09 0.06 0.25 0.2 0.24 0.15 0.01 Required Calculate the following: show all calculation as your proof. a) EOQ b) Re-order point c) Number of orders that would be placed in a year d) Total Relevant Cost e) Calculate the appropriate level of safety stock

a) Economic Order Quantity = 94.87

b) Re-order point = 3.21 units

c) Number of orders that would be placed in a year = 11.85

d) Total Relevant Cost = $699.07

e) The appropriate level of safety stock cannot be determined with the given information.

a) EOQ ( Economic Order Quantity ):

The EOQ formula is given by:

[tex]\[ EOQ = \sqrt{\frac{{2 \times D \times S}}{{H}}} \][/tex]

D = Annual demand (number of units sold annually) = 1,125 units

S = Cost to place an order = $4.00

H = Holding cost as a percentage of selling price = 14.4% of $100 (selling price)

Plugging in the values:

[tex]\[ EOQ = \sqrt{\frac{{2 \times 1,125 \times 4}}{{0.144 \times 100}}} \approx 94.87 \][/tex]

b) Re-order point:

The re-order point is the Lead Time Demand, calculated as:

[tex]\[ Lead\ Time\ Demand = Demand\ per\ day \times Lead\ time \][/tex]

Demand per day = Annual demand / (Number of working days per week × Number of weeks per year)

Lead time = 1 week

[tex]Demand per day = 1,125 / (7 \times 50) \approx 3.21\ units\ per\ day[/tex]

Lead Time Demand [tex]= 3.21 \times 1 = 3.21\ units[/tex]

c) Number of orders that would be placed in a year:

Number of orders = Annual demand / EOQ

Number of orders = [tex]1,125 / 94.87 \approx 11.85\ (rounded\ to\ 2\ decimal\ places)[/tex]

d) Total Relevant Cost :

Total Relevant Cost = Ordering Cost + Holding Cost

Ordering Cost = [tex]Cost to place an order \times Number of orders per year[/tex]

Holding Cost = [tex]Holding cost percentage \times Unit cost \times Average inventory[/tex]

Ordering Cost = [tex]4.00 \times 12 = 48.00[/tex]

Holding Cost = [tex]0.144 \times $100 \times (94.87 / 2) = $651.07[/tex]

Total Relevant Cost =[tex]$48.00 + $651.07 \approx $699.07\ (rounded\ to\ 2\ decimal\ places)[/tex]

e) Calculate the appropriate level of safety stock :

Since the standard deviation of demand during lead time is not provided, it's not possible to calculate the appropriate level of safety stock with the given information.

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Using state level data, a researcher wishes to examine the relationship between the median rent paid (RENT) as a function of median house values (MDHOUSE in $1,000). The percentage of the state population living in an urban area (PCTURBAN) is used as an additional control. a. The least squares estimates of the model arc in column c. In column (4) the least squares residuals (VHAT) from the regression in column (2) are added as a regressor to the basic regression. The estimates are obtained using least squares. Using the model in column (4), test if MDHOUSE is endogenous. What does the test result indicate about the results in (1)? d. Column (5) shows IV/2SLS estimates using the instruments listed in part (b). What differences do you observe between these results and the least squares results in column (1)? in particular, compare the estimates, standard errors and t-statistics to those in column (1). e. In column (6) the residuals from the estimation in (5) called EHAT are regressed upon the variables shown. What information is contained in these results? ((Any answers to questions d and e specifically?))

a. The test for endogeneity in the model is conducted by adding the least squares residuals (VHAT) from the regression in column (2) as a regressor to the basic regression in column (4). d. In column (5), the IV/2SLS estimates are obtained using instrumental variables listed in part (b).e. The regression of the residuals (EHAT) from the IV/2SLS estimation in column (5) onto the variables shown provides information about the validity and effectiveness of the instrumental variables used in the model.

a. The test for endogeneity in the model is conducted by adding the least squares residuals (VHAT) from the regression in column (2) as a regressor to the basic regression in column (4). If the coefficient of VHAT is statistically significant and different from zero, it suggests that there is endogeneity present in the model, indicating a potential issue with the least squares estimates in column (1).

d. In column (5), the IV/2SLS estimates are obtained using instrumental variables listed in part (b). A comparison between the results in column (5) and column (1) reveals differences in the estimates, standard errors, and t-statistics. The IV/2SLS estimates are aimed at addressing the potential endogeneity issue in the model by using instruments that are expected to be exogenous to the equation. These results provide an alternative estimation method that can potentially yield more reliable and unbiased estimates compared to the least squares estimates.

e. The regression of the residuals (EHAT) from the IV/2SLS estimation in column (5) onto the variables shown provides information about the validity and effectiveness of the instrumental variables used in the model. If the coefficients of the variables are statistically insignificant and close to zero, it suggests that the instruments are valid and not correlated with the residuals, supporting the reliability of the IV/2SLS estimates obtained in column (5). On the other hand, if the coefficients are statistically significant, it indicates a potential problem with the instruments and casts doubt on the validity of the IV/2SLS results.

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When reading the Cash Inflows from Operating Activities portion of the statement of cash flows using the direct method, you would expect to find which of the following? I. Collections from customers II. Interest and dividends collected III. Other operating receipts IV. Receipts from shareholders I, II, and III I, II, III, and IV I I and IV

When examining the Cash Inflows from Operating Activities portion of the statement of cash flows using the direct method, one would expect to find collections from customers , interest and dividends collected, and other operating receipts, but not receipts from stockholders. Therefore, the correct answer is A.

I. The cash inflows from operating activities should detail the amount of cash generated from the sale of products or services, among other operating activities. The collections from customers will be a significant component of the cash inflow from operating activities because it is the company's primary revenue source. The cash received from the interest and dividends from various investments made by the company would also be accounted for. When a company has non-operating income, such as an insurance claim, it may be included in this section. However, cash inflows from stockholders do not typically fall under cash inflows from operating activities because stockholders are not a source of revenue. Therefore, it is unlikely that cash inflows from stockholders will appear on the statement of cash flows using the direct method.

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When reading the Cash Inflows from Operating Activities portion of the statement of cash flows using the direct method, you would expect to find which of the following? I. Collections from customers II. Interest and dividends collected III. Other operating receipts IV. Receipts from stockholders

A. I B. I and IV C. I, II, and III D. I, II, III, and IV

A country does not trade with any other country. Its GDP is £20 billion and its government spending £2 billion. Each year the £8 billion of taxation is collected. The countries public saving amounts to £2 billion. What is the consumption and investment in the country?

In the absence of any trade, GDP can be measured by either of the three methods- by measuring the sum of all incomes earned, or by measuring the sum of all expenditures on final goods and services, or by measuring the value-added in all the stages of production.

The expenditure method is being used here. GDP = Consumption + Investment + Government Spending + Net Exports. Net Exports in this case is zero as there is no trade with other countries. So, the GDP can be calculated as follows, GDP = C + I + G + NX20 = C + I + 2 + 0.02 billion pounds of public saving are kept in the country and so the sum of public and private savings must equal the investment . Thus, the Investment would be the sum of private saving and public saving which is 2+2=4 billion pounds. Therefore, the consumption and investment in the country are £14 billion and £4 billion respectively.

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