Hypothesis Testing Calculator
| $H_o$: | |||
| $H_a$: | μ | ≠ | μ₀ |
| $n$ | = | $\bar{x}$ | = | = |
| $\text{Test Statistic: }$ | = |
| $\text{Degrees of Freedom: } $ | $df$ | = |
| $ \text{Level of Significance: } $ | $\alpha$ | = |
Type II Error
| $H_o$: | $\mu$ | ||
| $H_a$: | $\mu$ | ≠ | $\mu_0$ |
| $n$ | = | σ | = | $\mu$ | = |
| $\text{Level of Significance: }$ | $\alpha$ | = |
The first step in hypothesis testing is to calculate the test statistic. The formula for the test statistic depends on whether the population standard deviation (σ) is known or unknown. If σ is known, our hypothesis test is known as a z test and we use the z distribution. If σ is unknown, our hypothesis test is known as a t test and we use the t distribution. Use of the t distribution relies on the degrees of freedom, which is equal to the sample size minus one. Furthermore, if the population standard deviation σ is unknown, the sample standard deviation s is used instead. To switch from σ known to σ unknown, click on $\boxed{\sigma}$ and select $\boxed{s}$ in the Hypothesis Testing Calculator.
| $\sigma$ Known | $\sigma$ Unknown | |
| Test Statistic | $ z = \dfrac{\bar{x}-\mu_0}{\sigma/\sqrt{{\color{Black} n}}} $ | $ t = \dfrac{\bar{x}-\mu_0}{s/\sqrt{n}} $ |
Next, the test statistic is used to conduct the test using either the p-value approach or critical value approach. The particular steps taken in each approach largely depend on the form of the hypothesis test: lower tail, upper tail or two-tailed. The form can easily be identified by looking at the alternative hypothesis (H a ). If there is a less than sign in the alternative hypothesis then it is a lower tail test, greater than sign is an upper tail test and inequality is a two-tailed test. To switch from a lower tail test to an upper tail or two-tailed test, click on $\boxed{\geq}$ and select $\boxed{\leq}$ or $\boxed{=}$, respectively.
| Lower Tail Test | Upper Tail Test | Two-Tailed Test |
| $H_0 \colon \mu \geq \mu_0$ | $H_0 \colon \mu \leq \mu_0$ | $H_0 \colon \mu = \mu_0$ |
| $H_a \colon \mu | $H_a \colon \mu \neq \mu_0$ |
In the p-value approach, the test statistic is used to calculate a p-value. If the test is a lower tail test, the p-value is the probability of getting a value for the test statistic at least as small as the value from the sample. If the test is an upper tail test, the p-value is the probability of getting a value for the test statistic at least as large as the value from the sample. In a two-tailed test, the p-value is the probability of getting a value for the test statistic at least as unlikely as the value from the sample.
To test the hypothesis in the p-value approach, compare the p-value to the level of significance. If the p-value is less than or equal to the level of signifance, reject the null hypothesis. If the p-value is greater than the level of significance, do not reject the null hypothesis. This method remains unchanged regardless of whether it's a lower tail, upper tail or two-tailed test. To change the level of significance, click on $\boxed{.05}$. Note that if the test statistic is given, you can calculate the p-value from the test statistic by clicking on the switch symbol twice.
In the critical value approach, the level of significance ($\alpha$) is used to calculate the critical value. In a lower tail test, the critical value is the value of the test statistic providing an area of $\alpha$ in the lower tail of the sampling distribution of the test statistic. In an upper tail test, the critical value is the value of the test statistic providing an area of $\alpha$ in the upper tail of the sampling distribution of the test statistic. In a two-tailed test, the critical values are the values of the test statistic providing areas of $\alpha / 2$ in the lower and upper tail of the sampling distribution of the test statistic.
To test the hypothesis in the critical value approach, compare the critical value to the test statistic. Unlike the p-value approach, the method we use to decide whether to reject the null hypothesis depends on the form of the hypothesis test. In a lower tail test, if the test statistic is less than or equal to the critical value, reject the null hypothesis. In an upper tail test, if the test statistic is greater than or equal to the critical value, reject the null hypothesis. In a two-tailed test, if the test statistic is less than or equal the lower critical value or greater than or equal to the upper critical value, reject the null hypothesis.
| Lower Tail Test | Upper Tail Test | Two-Tailed Test |
| If $z \leq -z_\alpha$, reject $H_0$. | If $z \geq z_\alpha$, reject $H_0$. | If $z \leq -z_{\alpha/2}$ or $z \geq z_{\alpha/2}$, reject $H_0$. |
| If $t \leq -t_\alpha$, reject $H_0$. | If $t \geq t_\alpha$, reject $H_0$. | If $t \leq -t_{\alpha/2}$ or $t \geq t_{\alpha/2}$, reject $H_0$. |
When conducting a hypothesis test, there is always a chance that you come to the wrong conclusion. There are two types of errors you can make: Type I Error and Type II Error. A Type I Error is committed if you reject the null hypothesis when the null hypothesis is true. Ideally, we'd like to accept the null hypothesis when the null hypothesis is true. A Type II Error is committed if you accept the null hypothesis when the alternative hypothesis is true. Ideally, we'd like to reject the null hypothesis when the alternative hypothesis is true.
| Condition | ||||
| $H_0$ True | $H_a$ True | |||
| Conclusion | Accept $H_0$ | Correct | Type II Error | |
| Reject $H_0$ | Type I Error | Correct | ||
Hypothesis testing is closely related to the statistical area of confidence intervals. If the hypothesized value of the population mean is outside of the confidence interval, we can reject the null hypothesis. Confidence intervals can be found using the Confidence Interval Calculator . The calculator on this page does hypothesis tests for one population mean. Sometimes we're interest in hypothesis tests about two population means. These can be solved using the Two Population Calculator . The probability of a Type II Error can be calculated by clicking on the link at the bottom of the page.
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T-Test calculator
The Student's t-test is used to determine if means of two data sets differ significantly. This calculator will generate a step by step explanation on how to apply t – test.
Enter your z score value, and then press the button. Additional Z Statistic Calculators If you're interested in using the z statistic for hypothesis testing and the like, then we have a number of other calculators that might help you. Z-Test Calculator for a Single Sample Z-Test Calculator for 2 Population Proportions Z Score Calculator for a Single Raw Value (Includes z from p )  Two Sample t test calculatorInstructions: Use this calculator to work on a two-samples t-test, showing all the steps. In order to run the test, you need two provide two independent samples in the spreadsheet below. You can either type the data or simply paste them from Excel.  Two-sample t-test calculatorThis calculator will allow you to get all the details and steps related to the calculation of a two-sample t-test. The process for conducting a t-test is relatively simple, but it requires lots of calculations often times, which will be shown to you in detail by this calculator. The first step in using this calculator is to use the spreadsheet in which you need to either type or paste the data. You can have your data originally in Excel and then paste it in, no problem. After you type or paste the data, all you need to do is to click on "Calculate" to get all the steps shown. There are lots of subtleties involved in the process of conducting a t-test. There are certain distribution assumptions that need to be met, it needs to be assessed whether or not the population standard deviation can be assumed to be equal . Once the assumption requirements are cleared, we can proceed with the test statistic calculation.  Independent t-test Calculator with SamplesUsually there are two different forms that can lead to calculating an independent t-test. You can either have two samples, or you can have the data already summarized. For the latter, use this independent t-test calculator with summarized data . For the case of two samples, you will first need to conduct descriptive statistics calculations in order to get a summary of the provided independent samples. Steps for running a independent t-test
Why do we need to test for the equality of population variances? This is because there is the need to find the standard error for the test, and it turns out that the optimal choice for the standard error depends on whether the population standard deviations are equal or not. That is a rather technical topic, but in layman terms, if the population variances are equal, then the best choice is to basically pool the available sample variances to get a good standard error estimate. But if they are not equal, things get a bit more complicated, and some technical corrections are needed, which is what you see reflected in the fact that the formula used is different, and the degrees of freedom are different too. What is the t-value in a 2 sample test?The formula used for the independent samples t-test will depend upon whether or not the population variances are assumed to be equal. If they are assumed to be unequal, the formula used is But, if the population variances are assumed to be equal, you then need to use the following formula: Equality of Population VariancesWhen to assume equality of population variances? There is a formal test, which is the F-test for equality of variances, which is conducted by this calculator if you select the option. Sometimes, different rules of thumb are used, like taking the highest of sample variance, divide by the lowest sample variance and assume that the population variances are equal if this ratio is less than 3, or another rule like that. That is not a completely bad idea, but if you really need to know, it is best to run a formal test. What are the steps for computing the t-test formula
The number of degrees of freedom when equal population variances are assumed is \(df = n_1 + n_2\), where \(n_1\) and \(n_2\) are the corresponding sample sizes . Now for unequal variances, the calculation of degrees of freedoms is a lot more complicated. Is this a t-test calculator with steps?Yes! This calculator will show you all the steps of the way, from the calculation of descriptive statistics, to the testing for equality of variances (if required) to the use of the proper t-test formula to the discussion and conclusions. Why is this test statistic calculator useful? Time! You will save lots of time because an independent samples t-test requires a whole lot of calculations.  What is an example of a 2 sample t-test?Suppose that a teacher believes that the average height of eighth graders for two different schools. There is a sample of n = 10 kids for each school, for which their sample heights (in inches) are available: School 1: 60, 62, 59, 63, 65, 64, 68, 67, 61, 60 School 1: 60, 61, 61, 61, 60, 59, 59, 60, 60, 59 Is there enough evidence to claim that the population mean heights for two schools are different, at the 0.05 significance level? Solution: The following information sample information has been provided:
In order to conduct a two-independent samples t-test, we need to compute descriptive statistics of the samples:
Summarizing, the following descriptive statistics will be used in the calculation of the t-statistic: The following information has been provided:
(1) Null and Alternative Hypotheses The following null and alternative hypotheses need to be tested: This corresponds to a two-tailed test, for which a t-test for two population means, with two independent samples, with unknown population standard deviations will be used. Testing for Equality of Variances A F-test is used to test for the equality of variances. The following F-ratio is obtained: The critical values are \(F_L = 0.248\) and \(F_U = 4.026\), and since \(F = 14.15\), then the null hypothesis of equal variances is rejected. (2) Rejection Region Based on the information provided, the significance level is \(\alpha = 0.05\), and the degrees of freedom are \(df = 10.266\). In fact, the degrees of freedom are computed as follows, assuming that the population variances are unequal: Hence, it is found that the critical value for this two-tailed test is \(t_c = 2.22\), for \(\alpha = 0.05\) and \(df = 10.266\). The rejection region for this two-tailed test is \(R = \{t: |t| > 2.22\}\). (3) Test Statistics Since it is assumed that the population variances are unequal, the t-statistic is computed as follows: (4) Decision about the null hypothesis Since it is observed that \(|t| = 2.886 > t_c = 2.22\), it is then concluded that the null hypothesis is rejected. Using the P-value approach: The p-value is \(p = 0.0158\), and since \(p = 0.0158 < 0.05\), it is concluded that the null hypothesis is rejected. (5) Conclusion It is concluded that the null hypothesis Ho is rejected. Therefore, there is enough evidence to claim that the population mean \(\mu_1\) is different than \(\mu_2\), at the \(\alpha = 0.05\) significance level. Confidence Interval The 95% confidence interval is \(0.669 < \mu < 5.131\). Graphically  Other statistical tests of interestThere is an abundance of related statistical tests that you can use. You can try for example this paired t-test calculator . Also you can you this t-test for two samples when you have summarized sample data instead. In that case, the sample data provided is usually the sample means , sample standard deviations and sample sizes. Other type of t-test calculators include the t-test for one sample . For different types of statistics, you can try this ANOVA calculator , which is similar to the t-test only that with ANOVA you can compare more than 2 groups. Related Calculators log in to your accountReset password. Critical Value CalculatorUse this calculator for critical values to easily convert a significance level to its corresponding Z value, T score, F-score, or Chi-square value. Outputs the critical region as well. The tool supports one-tailed and two-tailed significance tests / probability values. Related calculators
Using the critical value calculatorIf you want to perform a statistical test of significance (a.k.a. significance test, statistical significance test), determining the value of the test statistic corresponding to the desired significance level is necessary. You need to know the desired error probability ( p-value threshold , common values are 0.05, 0.01, 0.001) corresponding to the significance level of the test. If you know the significance level in percentages, simply subtract it from 100%. For example, 95% significance results in a probability of 100%-95% = 5% = 0.05 . Then you need to know the shape of the error distribution of the statistic of interest (not to be mistaken with the distribution of the underlying data!) . Our critical value calculator supports statistics which are either:
Then, for distributions other than the normal one (Z), you need to know the degrees of freedom . For the F statistic there are two separate degrees of freedom - one for the numerator and one for the denominator. Finally, to determine a critical region, one needs to know whether they are testing a point null versus a composite alternative (on both sides) or a composite null versus (covering one side of the distribution) a composite alternative (covering the other). Basically, it comes down to whether the inference is going to contain claims regarding the direction of the effect or not. Should one want to claim anything about the direction of the effect, the corresponding null hypothesis is direction as well (one-sided hypothesis). Depending on the type of test - one-tailed or two-tailed, the calculator will output the critical value or values and the corresponding critical region. For one-sided tests it will output both possible regions, whereas for a two-sided test it will output the union of the two critical regions on the opposite sides of the distribution. What is a critical value?A critical value (or values) is a point on the support of an error distribution which bounds a critical region from above or below. If the statistics falls below or above a critical value (depending on the type of hypothesis, but it has to fall inside the critical region) then a test is declared statistically significant at the corresponding significance level. For example, in a two-tailed Z test with critical values -1.96 and 1.96 (corresponding to 0.05 significance level) the critical regions are from -∞ to -1.96 and from 1.96 to +∞. Therefore, if the statistic falls below -1.96 or above 1.96, the null hypothesis test is statistically significant. You can think of the critical value as a cutoff point beyond which events are considered rare enough to count as evidence against the specified null hypothesis. It is a value achieved by a distance function with probability equal to or greater than the significance level under the specified null hypothesis. In an error-probabilistic framework, a proper distance function based on a test statistic takes the generic form [1] :  X (read "X bar") is the arithmetic mean of the population baseline or the control, μ 0 is the observed mean / treatment group mean, while σ x is the standard error of the mean (SEM, or standard deviation of the error of the mean). Here is how it looks in practice when the error is normally distributed (Z distribution) with a one-tailed null and alternative hypotheses and a significance level α set to 0.05:  And here is the same significance level when applied to a point null and a two-tailed alternative hypothesis:  The distance function would vary depending on the distribution of the error: Z, T, F, or Chi-square (X 2 ). The calculation of a particular critical value based on a supplied probability and error distribution is simply a matter of calculating the inverse cumulative probability density function (inverse CPDF) of the respective distribution. This can be a difficult task, most notably for the T distribution [2] . T critical value calculationThe T-distribution is often preferred in the social sciences, psychiatry, economics, and other sciences where low sample sizes are a common occurrence. Certain clinical studies also fall under this umbrella. This stems from the fact that for sample sizes over 30 it is practically equivalent to the normal distribution which is easier to work with. It was proposed by William Gosset, a.k.a. Student, in 1908 [3] , which is why it is also referred to as "Student's T distribution". To find the critical t value, one needs to compute the inverse cumulative PDF of the T distribution. To do that, the significance level and the degrees of freedom need to be known. The degrees of freedom represent the number of values in the final calculation of a statistic that are free to vary whilst the statistic remains fixed at a certain value. It should be noted that there is not, in fact, a single T-distribution, but there are infinitely many T-distributions, each with a different level of degrees of freedom. Below are some key values of the T-distribution with 1 degree of freedom, assuming a one-tailed T test is to be performed. These are often used as critical values to define rejection regions in hypothesis testing.
Z critical value calculationThe Z-score is a statistic showing how many standard deviations away from the normal, usually the mean, a given observation is. It is often called just a standard score, z-value, normal score, and standardized variable. A Z critical value is just a particular cutoff in the error distribution of a normally-distributed statistic. Z critical values are computed by using the inverse cumulative probability density function of the standard normal distribution with a mean (μ) of zero and standard deviation (σ) of one. Below are some commonly encountered probability values (significance levels) and their corresponding Z values for the critical region, assuming a one-tailed hypothesis .
The critical region defined by each of these would span from the Z value to plus infinity for the right-tailed case, and from minus infinity to minus the Z critical value in the left-tailed case. Our calculator for critical value will both find the critical z value(s) and output the corresponding critical regions for you. Chi Square (Χ 2 ) critical value calculationChi square distributed errors are commonly encountered in goodness-of-fit tests and homogeneity tests, but also in tests for independence in contingency tables. Since the distribution is based on the squares of scores, it only contains positive values. Calculating the inverse cumulative PDF of the distribution is required in order to convert a desired probability (significance) to a chi square critical value. Just like the T and F distributions, there is a different chi square distribution corresponding to different degrees of freedom. Hence, to calculate a Χ 2 critical value one needs to supply the degrees of freedom for the statistic of interest. F critical value calculationF distributed errors are commonly encountered in analysis of variance (ANOVA), which is very common in the social sciences. The distribution, also referred to as the Fisher-Snedecor distribution, only contains positive values, similar to the Χ 2 one. Similar to the T distribution, there is no single F-distribution to speak of. A different F distribution is defined for each pair of degrees of freedom - one for the numerator and one for the denominator. Calculating the inverse cumulative PDF of the F distribution specified by the two degrees of freedom is required in order to convert a desired probability (significance) to a critical value. There is no simple solution to find a critical value of f and while there are tables, using a calculator is the preferred approach nowadays. References1 Mayo D.G., Spanos A. (2010) – "Error Statistics", in P. S. Bandyopadhyay & M. R. Forster (Eds.), Philosophy of Statistics, (7, 152–198). Handbook of the Philosophy of Science . The Netherlands: Elsevier. 2 Shaw T.W. (2006) – "Sampling Student's T distribution – use of the inverse cumulative distribution function", Journal of Computational Finance 9(4):37-73, DOI:10.21314/JCF.2006.150 3 "Student" [William Sealy Gosset] (1908) - "The probable error of a mean", Biometrika 6(1):1–25. DOI:10.1093/biomet/6.1.1 Cite this calculator & pageIf you'd like to cite this online calculator resource and information as provided on the page, you can use the following citation: Georgiev G.Z., "Critical Value Calculator" , [online] Available at: https://www.gigacalculator.com/calculators/critical-value-calculator.php URL [Accessed Date: 24 Aug, 2024]. Our statistical calculators have been featured in scientific papers and articles published in high-profile science journals by:  The author of this tool Statistical calculators Two Sample t-test: Definition, Formula, and ExampleA two sample t-test is used to determine whether or not two population means are equal. This tutorial explains the following:
Two Sample t-test: MotivationSuppose we want to know whether or not the mean weight between two different species of turtles is equal. Since there are thousands of turtles in each population, it would be too time-consuming and costly to go around and weigh each individual turtle. Instead, we might take a simple random sample of 15 turtles from each population and use the mean weight in each sample to determine if the mean weight is equal between the two populations:  However, it’s virtually guaranteed that the mean weight between the two samples will be at least a little different. The question is whether or not this difference is statistically significant . Fortunately, a two sample t-test allows us to answer this question. Two Sample t-test: FormulaA two-sample t-test always uses the following null hypothesis:
The alternative hypothesis can be either two-tailed, left-tailed, or right-tailed:
We use the following formula to calculate the test statistic t: Test statistic: ( x 1 – x 2 ) / s p (√ 1/n 1 + 1/n 2 ) where x 1 and x 2 are the sample means, n 1 and n 2 are the sample sizes, and where s p is calculated as: s p = √ (n 1 -1)s 1 2 + (n 2 -1)s 2 2 / (n 1 +n 2 -2) where s 1 2 and s 2 2 are the sample variances. If the p-value that corresponds to the test statistic t with (n 1 +n 2 -1) degrees of freedom is less than your chosen significance level (common choices are 0.10, 0.05, and 0.01) then you can reject the null hypothesis. Two Sample t-test: AssumptionsFor the results of a two sample t-test to be valid, the following assumptions should be met:
Two Sample t-test : ExampleSuppose we want to know whether or not the mean weight between two different species of turtles is equal. To test this, will perform a two sample t-test at significance level α = 0.05 using the following steps: Step 1: Gather the sample data. Suppose we collect a random sample of turtles from each population with the following information:
Step 2: Define the hypotheses. We will perform the two sample t-test with the following hypotheses:
Step 3: Calculate the test statistic t . First, we will calculate the pooled standard deviation s p : s p = √ (n 1 -1)s 1 2 + (n 2 -1)s 2 2 / (n 1 +n 2 -2) = √ (40-1)18.5 2 + (38-1)16.7 2 / (40+38-2) = 17.647 Next, we will calculate the test statistic t : t = ( x 1 – x 2 ) / s p (√ 1/n 1 + 1/n 2 ) = (300-305) / 17.647(√ 1/40 + 1/38 ) = -1.2508 Step 4: Calculate the p-value of the test statistic t . According to the T Score to P Value Calculator , the p-value associated with t = -1.2508 and degrees of freedom = n 1 +n 2 -2 = 40+38-2 = 76 is 0.21484 . Step 5: Draw a conclusion. Since this p-value is not less than our significance level α = 0.05, we fail to reject the null hypothesis. We do not have sufficient evidence to say that the mean weight of turtles between these two populations is different. Note: You can also perform this entire two sample t-test by simply using the Two Sample t-test Calculator . Additional ResourcesThe following tutorials explain how to perform a two-sample t-test using different statistical programs: How to Perform a Two Sample t-test in Excel How to Perform a Two Sample t-test in SPSS How to Perform a Two Sample t-test in Stata How to Perform a Two Sample t-test in R How to Perform a Two Sample t-test in Python How to Perform a Two Sample t-test on a TI-84 Calculator Featured Posts 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. 2 Replies to “Two Sample t-test: Definition, Formula, and Example”I like the detailed information and simplified in the way I can understand and relate easily. Thank you It seems a couple of parenthesis is missed at the pooled standard deviation formula. Under square root you have (n1-1)s12 + (n2-1)s22 / (n1+n2-2) but it should be [(n1-1)s12 + (n2-1)s22] / (n1+n2-2) I used square bracket Leave a Reply Cancel replyYour email address will not be published. Required fields are marked * Join the Statology CommunitySign up to receive Statology's exclusive study resource: 100 practice problems with step-by-step solutions. Plus, get our latest insights, tutorials, and data analysis tips straight to your inbox! By subscribing you accept Statology's Privacy Policy. Statistics TutorialDescriptive statistics, inferential statistics, stat reference, statistics - hypothesis testing a mean (two tailed). A population mean is an average of value a population. Hypothesis tests are used to check a claim about the size of that population mean. Hypothesis Testing a MeanThe following steps are used for a hypothesis test:
For example:
And we want to check the claim: "The average age of Nobel Prize winners when they received the prize is not 60" By taking a sample of 30 randomly selected Nobel Prize winners we could find that:
From this sample data we check the claim with the steps below. 1. Checking the ConditionsThe conditions for calculating a confidence interval for a proportion are:
A moderately large sample size, like 30, is typically large enough. In the example, the sample size was 30 and it was randomly selected, so the conditions are fulfilled. Note: Checking if the data is normally distributed can be done with specialized statistical tests. 2. Defining the ClaimsWe need to define a null hypothesis (\(H_{0}\)) and an alternative hypothesis (\(H_{1}\)) based on the claim we are checking. The claim was: In this case, the parameter is the mean age of Nobel Prize winners when they received the prize (\(\mu\)). The null and alternative hypothesis are then: Null hypothesis : The average age was 60. Alternative hypothesis : The average age is not 60. Which can be expressed with symbols as: \(H_{0}\): \(\mu = 60 \) \(H_{1}\): \(\mu \neq 60 \) This is a ' two-tailed ' test, because the alternative hypothesis claims that the proportion is different from the null hypothesis. If the data supports the alternative hypothesis, we reject the null hypothesis and accept the alternative hypothesis. Advertisement 3. Deciding the Significance LevelThe significance level (\(\alpha\)) is the uncertainty we accept when rejecting the null hypothesis in a hypothesis test. The significance level is a percentage probability of accidentally making the wrong conclusion. Typical significance levels are:
A lower significance level means that the evidence in the data needs to be stronger to reject the null hypothesis. There is no "correct" significance level - it only states the uncertainty of the conclusion. Note: A 5% significance level means that when we reject a null hypothesis: We expect to reject a true null hypothesis 5 out of 100 times. 4. Calculating the Test StatisticThe test statistic is used to decide the outcome of the hypothesis test. The test statistic is a standardized value calculated from the sample. The formula for the test statistic (TS) of a population mean is: \(\displaystyle \frac{\bar{x} - \mu}{s} \cdot \sqrt{n} \) \(\bar{x}-\mu\) is the difference between the sample mean (\(\bar{x}\)) and the claimed population mean (\(\mu\)). \(s\) is the sample standard deviation . \(n\) is the sample size. In our example: The claimed (\(H_{0}\)) population mean (\(\mu\)) was \( 60 \) The sample mean (\(\bar{x}\)) was \(62.1\) The sample standard deviation (\(s\)) was \(13.46\) The sample size (\(n\)) was \(30\) So the test statistic (TS) is then: \(\displaystyle \frac{62.1-60}{13.46} \cdot \sqrt{30} = \frac{2.1}{13.46} \cdot \sqrt{30} \approx 0.156 \cdot 5.477 = \underline{0.855}\) You can also calculate the test statistic using programming language functions: With Python use the scipy and math libraries to calculate the test statistic. With R use built-in math and statistics functions to calculate the test statistic. 5. ConcludingThere are two main approaches for making the conclusion of a hypothesis test:
Note: The two approaches are only different in how they present the conclusion. The Critical Value ApproachFor the critical value approach we need to find the critical value (CV) of the significance level (\(\alpha\)). For a population mean test, the critical value (CV) is a T-value from a student's t-distribution . This critical T-value (CV) defines the rejection region for the test. The rejection region is an area of probability in the tails of the standard normal distribution. Because the claim is that the population proportion is different from 60, the rejection region is split into both the left and right tail: The student's t-distribution is adjusted for the uncertainty from smaller samples. This adjustment is called degrees of freedom (df), which is the sample size \((n) - 1\) In this case the degrees of freedom (df) is: \(30 - 1 = \underline{29} \) Choosing a significance level (\(\alpha\)) of 0.05, or 5%, we can find the critical T-value from a T-table , or with a programming language function: Note: Because this is a two-tailed test the tail area (\(\alpha\)) needs to be split in half (divided by 2). With Python use the Scipy Stats library t.ppf() function find the T-Value for an \(\alpha\)/2 = 0.025 at 29 degrees of freedom (df). With R use the built-in qt() function to find the t-value for an \(\alpha\)/ = 0.025 at 29 degrees of freedom (df). Using either method we can find that the critical T-Value is \(\approx \underline{-2.045}\) For a two-tailed test we need to check if the test statistic (TS) is smaller than the negative critical value (-CV), or bigger than the positive critical value (CV). If the test statistic is smaller than the negative critical value, the test statistic is in the rejection region . If the test statistic is bigger than the positive critical value, the test statistic is in the rejection region . When the test statistic is in the rejection region, we reject the null hypothesis (\(H_{0}\)). Here, the test statistic (TS) was \(\approx \underline{0.855}\) and the critical value was \(\approx \underline{-2.045}\) Here is an illustration of this test in a graph: Since the test statistic is between the critical values we keep the null hypothesis. This means that the sample data does not support the alternative hypothesis. And we can summarize the conclusion stating: The sample data does not support the claim that "The average age of Nobel Prize winners when they received the prize is not 60" at a 5% significance level . The P-Value ApproachFor the P-value approach we need to find the P-value of the test statistic (TS). If the P-value is smaller than the significance level (\(\alpha\)), we reject the null hypothesis (\(H_{0}\)). The test statistic was found to be \( \approx \underline{0.855} \) For a population proportion test, the test statistic is a T-Value from a student's t-distribution . Because this is a two-tailed test, we need to find the P-value of a T-value bigger than 0.855 and multiply it by 2 . The student's t-distribution is adjusted according to degrees of freedom (df), which is the sample size \((30) - 1 = \underline{29}\) We can find the P-value using a T-table , or with a programming language function: With Python use the Scipy Stats library t.cdf() function find the P-value of a T-value bigger than 0.855 for a two tailed test at 29 degrees of freedom (df): With R use the built-in pt() function find the P-value of a T-Value bigger than 0.855 for a two tailed test at 29 degrees of freedom (df): Using either method we can find that the P-value is \(\approx \underline{0.3996}\) This tells us that the significance level (\(\alpha\)) would need to be smaller 0.3996, or 39.96%, to reject the null hypothesis. This P-value is bigger than any of the common significance levels (10%, 5%, 1%). So the null hypothesis is kept at all of these significance levels. The sample data does not support the claim that "The average age of Nobel Prize winners when they received the prize is not 60" at a 10%, 5%, or 1% significance level . Calculating a P-Value for a Hypothesis Test with ProgrammingMany programming languages can calculate the P-value to decide outcome of a hypothesis test. Using software and programming to calculate statistics is more common for bigger sets of data, as calculating manually becomes difficult. The P-value calculated here will tell us the lowest possible significance level where the null-hypothesis can be rejected. With Python use the scipy and math libraries to calculate the P-value for a two tailed hypothesis test for a mean. Here, the sample size is 30, the sample mean is 62.1, the sample standard deviation is 13.46, and the test is for a mean different from 60. With R use built-in math and statistics functions find the P-value for a two tailed hypothesis test for a mean. Left-Tailed and Two-Tailed TestsThis was an example of a left tailed test, where the alternative hypothesis claimed that parameter is smaller than the null hypothesis claim. You can check out an equivalent step-by-step guide for other types here:
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Calculator: One-Sample t-Test
One-Sample t-Test Calculator
This calculator will conduct a complete one-sample t-test, given the sample mean, the sample size, the hypothesized mean, and the sample standard deviation. The results generated by the calculator include the t-statistic, the degrees of freedom, the critical t-values for both one-tailed (directional) and two-tailed (non-directional) hypotheses, and the one-tailed and two-tailed probability values associated with the test. Please enter the necessary parameter values, and then click 'Calculate'.
| Sample mean (x): | ||
| Sample size: | ||
| Sample standard deviation: |
p-value Calculator
What is p-value, how do i calculate p-value from test statistic, how to interpret p-value, how to use the p-value calculator to find p-value from test statistic, how do i find p-value from z-score, how do i find p-value from t, p-value from chi-square score (χ² score), p-value from f-score.
Welcome to our p-value calculator! You will never again have to wonder how to find the p-value, as here you can determine the one-sided and two-sided p-values from test statistics, following all the most popular distributions: normal, t-Student, chi-squared, and Snedecor's F.
P-values appear all over science, yet many people find the concept a bit intimidating. Don't worry – in this article, we will explain not only what the p-value is but also how to interpret p-values correctly . Have you ever been curious about how to calculate the p-value by hand? We provide you with all the necessary formulae as well!
🙋 If you want to revise some basics from statistics, our normal distribution calculator is an excellent place to start.
Formally, the p-value is the probability that the test statistic will produce values at least as extreme as the value it produced for your sample . It is crucial to remember that this probability is calculated under the assumption that the null hypothesis H 0 is true !
More intuitively, p-value answers the question:
Assuming that I live in a world where the null hypothesis holds, how probable is it that, for another sample, the test I'm performing will generate a value at least as extreme as the one I observed for the sample I already have?
It is the alternative hypothesis that determines what "extreme" actually means , so the p-value depends on the alternative hypothesis that you state: left-tailed, right-tailed, or two-tailed. In the formulas below, S stands for a test statistic, x for the value it produced for a given sample, and Pr(event | H 0 ) is the probability of an event, calculated under the assumption that H 0 is true:
Left-tailed test: p-value = Pr(S ≤ x | H 0 )
Right-tailed test: p-value = Pr(S ≥ x | H 0 )
Two-tailed test:
p-value = 2 × min{Pr(S ≤ x | H 0 ), Pr(S ≥ x | H 0 )}
(By min{a,b} , we denote the smaller number out of a and b .)
If the distribution of the test statistic under H 0 is symmetric about 0 , then: p-value = 2 × Pr(S ≥ |x| | H 0 )
or, equivalently: p-value = 2 × Pr(S ≤ -|x| | H 0 )
As a picture is worth a thousand words, let us illustrate these definitions. Here, we use the fact that the probability can be neatly depicted as the area under the density curve for a given distribution. We give two sets of pictures: one for a symmetric distribution and the other for a skewed (non-symmetric) distribution.
- Symmetric case: normal distribution:
- Non-symmetric case: chi-squared distribution:
In the last picture (two-tailed p-value for skewed distribution), the area of the left-hand side is equal to the area of the right-hand side.
To determine the p-value, you need to know the distribution of your test statistic under the assumption that the null hypothesis is true . Then, with the help of the cumulative distribution function ( cdf ) of this distribution, we can express the probability of the test statistics being at least as extreme as its value x for the sample:
Left-tailed test:
p-value = cdf(x) .
Right-tailed test:
p-value = 1 - cdf(x) .
p-value = 2 × min{cdf(x) , 1 - cdf(x)} .
If the distribution of the test statistic under H 0 is symmetric about 0 , then a two-sided p-value can be simplified to p-value = 2 × cdf(-|x|) , or, equivalently, as p-value = 2 - 2 × cdf(|x|) .
The probability distributions that are most widespread in hypothesis testing tend to have complicated cdf formulae, and finding the p-value by hand may not be possible. You'll likely need to resort to a computer or to a statistical table, where people have gathered approximate cdf values.
Well, you now know how to calculate the p-value, but… why do you need to calculate this number in the first place? In hypothesis testing, the p-value approach is an alternative to the critical value approach . Recall that the latter requires researchers to pre-set the significance level, α, which is the probability of rejecting the null hypothesis when it is true (so of type I error ). Once you have your p-value, you just need to compare it with any given α to quickly decide whether or not to reject the null hypothesis at that significance level, α. For details, check the next section, where we explain how to interpret p-values.
As we have mentioned above, the p-value is the answer to the following question:
What does that mean for you? Well, you've got two options:
- A high p-value means that your data is highly compatible with the null hypothesis; and
- A small p-value provides evidence against the null hypothesis , as it means that your result would be very improbable if the null hypothesis were true.
However, it may happen that the null hypothesis is true, but your sample is highly unusual! For example, imagine we studied the effect of a new drug and got a p-value of 0.03 . This means that in 3% of similar studies, random chance alone would still be able to produce the value of the test statistic that we obtained, or a value even more extreme, even if the drug had no effect at all!
The question "what is p-value" can also be answered as follows: p-value is the smallest level of significance at which the null hypothesis would be rejected. So, if you now want to make a decision on the null hypothesis at some significance level α , just compare your p-value with α :
- If p-value ≤ α , then you reject the null hypothesis and accept the alternative hypothesis; and
- If p-value ≥ α , then you don't have enough evidence to reject the null hypothesis.
Obviously, the fate of the null hypothesis depends on α . For instance, if the p-value was 0.03 , we would reject the null hypothesis at a significance level of 0.05 , but not at a level of 0.01 . That's why the significance level should be stated in advance and not adapted conveniently after the p-value has been established! A significance level of 0.05 is the most common value, but there's nothing magical about it. Here, you can see what too strong a faith in the 0.05 threshold can lead to. It's always best to report the p-value, and allow the reader to make their own conclusions.
Also, bear in mind that subject area expertise (and common reason) is crucial. Otherwise, mindlessly applying statistical principles, you can easily arrive at statistically significant, despite the conclusion being 100% untrue.
As our p-value calculator is here at your service, you no longer need to wonder how to find p-value from all those complicated test statistics! Here are the steps you need to follow:
Pick the alternative hypothesis : two-tailed, right-tailed, or left-tailed.
Tell us the distribution of your test statistic under the null hypothesis: is it N(0,1), t-Student, chi-squared, or Snedecor's F? If you are unsure, check the sections below, as they are devoted to these distributions.
If needed, specify the degrees of freedom of the test statistic's distribution.
Enter the value of test statistic computed for your data sample.
Our calculator determines the p-value from the test statistic and provides the decision to be made about the null hypothesis. The standard significance level is 0.05 by default.
Go to the advanced mode if you need to increase the precision with which the calculations are performed or change the significance level .
In terms of the cumulative distribution function (cdf) of the standard normal distribution, which is traditionally denoted by Φ , the p-value is given by the following formulae:
Left-tailed z-test:
p-value = Φ(Z score )
Right-tailed z-test:
p-value = 1 - Φ(Z score )
Two-tailed z-test:
p-value = 2 × Φ(−|Z score |)
p-value = 2 - 2 × Φ(|Z score |)
🙋 To learn more about Z-tests, head to Omni's Z-test calculator .
We use the Z-score if the test statistic approximately follows the standard normal distribution N(0,1) . Thanks to the central limit theorem, you can count on the approximation if you have a large sample (say at least 50 data points) and treat your distribution as normal.
A Z-test most often refers to testing the population mean , or the difference between two population means, in particular between two proportions. You can also find Z-tests in maximum likelihood estimations.
The p-value from the t-score is given by the following formulae, in which cdf t,d stands for the cumulative distribution function of the t-Student distribution with d degrees of freedom:
Left-tailed t-test:
p-value = cdf t,d (t score )
Right-tailed t-test:
p-value = 1 - cdf t,d (t score )
Two-tailed t-test:
p-value = 2 × cdf t,d (−|t score |)
p-value = 2 - 2 × cdf t,d (|t score |)
Use the t-score option if your test statistic follows the t-Student distribution . This distribution has a shape similar to N(0,1) (bell-shaped and symmetric) but has heavier tails – the exact shape depends on the parameter called the degrees of freedom . If the number of degrees of freedom is large (>30), which generically happens for large samples, the t-Student distribution is practically indistinguishable from the normal distribution N(0,1).
The most common t-tests are those for population means with an unknown population standard deviation, or for the difference between means of two populations , with either equal or unequal yet unknown population standard deviations. There's also a t-test for paired (dependent) samples .
🙋 To get more insights into t-statistics, we recommend using our t-test calculator .
Use the χ²-score option when performing a test in which the test statistic follows the χ²-distribution .
This distribution arises if, for example, you take the sum of squared variables, each following the normal distribution N(0,1). Remember to check the number of degrees of freedom of the χ²-distribution of your test statistic!
How to find the p-value from chi-square-score ? You can do it with the help of the following formulae, in which cdf χ²,d denotes the cumulative distribution function of the χ²-distribution with d degrees of freedom:
Left-tailed χ²-test:
p-value = cdf χ²,d (χ² score )
Right-tailed χ²-test:
p-value = 1 - cdf χ²,d (χ² score )
Remember that χ²-tests for goodness-of-fit and independence are right-tailed tests! (see below)
Two-tailed χ²-test:
p-value = 2 × min{cdf χ²,d (χ² score ), 1 - cdf χ²,d (χ² score )}
(By min{a,b} , we denote the smaller of the numbers a and b .)
The most popular tests which lead to a χ²-score are the following:
Testing whether the variance of normally distributed data has some pre-determined value. In this case, the test statistic has the χ²-distribution with n - 1 degrees of freedom, where n is the sample size. This can be a one-tailed or two-tailed test .
Goodness-of-fit test checks whether the empirical (sample) distribution agrees with some expected probability distribution. In this case, the test statistic follows the χ²-distribution with k - 1 degrees of freedom, where k is the number of classes into which the sample is divided. This is a right-tailed test .
Independence test is used to determine if there is a statistically significant relationship between two variables. In this case, its test statistic is based on the contingency table and follows the χ²-distribution with (r - 1)(c - 1) degrees of freedom, where r is the number of rows, and c is the number of columns in this contingency table. This also is a right-tailed test .
Finally, the F-score option should be used when you perform a test in which the test statistic follows the F-distribution , also known as the Fisher–Snedecor distribution. The exact shape of an F-distribution depends on two degrees of freedom .
To see where those degrees of freedom come from, consider the independent random variables X and Y , which both follow the χ²-distributions with d 1 and d 2 degrees of freedom, respectively. In that case, the ratio (X/d 1 )/(Y/d 2 ) follows the F-distribution, with (d 1 , d 2 ) -degrees of freedom. For this reason, the two parameters d 1 and d 2 are also called the numerator and denominator degrees of freedom .
The p-value from F-score is given by the following formulae, where we let cdf F,d1,d2 denote the cumulative distribution function of the F-distribution, with (d 1 , d 2 ) -degrees of freedom:
Left-tailed F-test:
p-value = cdf F,d1,d2 (F score )
Right-tailed F-test:
p-value = 1 - cdf F,d1,d2 (F score )
Two-tailed F-test:
p-value = 2 × min{cdf F,d1,d2 (F score ), 1 - cdf F,d1,d2 (F score )}
Below we list the most important tests that produce F-scores. All of them are right-tailed tests .
A test for the equality of variances in two normally distributed populations . Its test statistic follows the F-distribution with (n - 1, m - 1) -degrees of freedom, where n and m are the respective sample sizes.
ANOVA is used to test the equality of means in three or more groups that come from normally distributed populations with equal variances. We arrive at the F-distribution with (k - 1, n - k) -degrees of freedom, where k is the number of groups, and n is the total sample size (in all groups together).
A test for overall significance of regression analysis . The test statistic has an F-distribution with (k - 1, n - k) -degrees of freedom, where n is the sample size, and k is the number of variables (including the intercept).
With the presence of the linear relationship having been established in your data sample with the above test, you can calculate the coefficient of determination, R 2 , which indicates the strength of this relationship . You can do it by hand or use our coefficient of determination calculator .
A test to compare two nested regression models . The test statistic follows the F-distribution with (k 2 - k 1 , n - k 2 ) -degrees of freedom, where k 1 and k 2 are the numbers of variables in the smaller and bigger models, respectively, and n is the sample size.
You may notice that the F-test of an overall significance is a particular form of the F-test for comparing two nested models: it tests whether our model does significantly better than the model with no predictors (i.e., the intercept-only model).
Can p-value be negative?
No, the p-value cannot be negative. This is because probabilities cannot be negative, and the p-value is the probability of the test statistic satisfying certain conditions.
What does a high p-value mean?
A high p-value means that under the null hypothesis, there's a high probability that for another sample, the test statistic will generate a value at least as extreme as the one observed in the sample you already have. A high p-value doesn't allow you to reject the null hypothesis.
What does a low p-value mean?
A low p-value means that under the null hypothesis, there's little probability that for another sample, the test statistic will generate a value at least as extreme as the one observed for the sample you already have. A low p-value is evidence in favor of the alternative hypothesis – it allows you to reject the null hypothesis.
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7.4: Hypothesis Tests for a Single Population Mean
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- Page ID 49002
- Hannah Seidler-Wright
- Chaffey College
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In previous lessons, we have learned that there are two fundamental forms of statistical inference: confidence intervals and hypothesis tests. In the last section, we used confidence intervals to estimate a single population mean. We will now apply the four step hypothesis testing process to test hypotheses about the value of a single population mean. To explore this concept, we will examine examples of water contamination across the US.
The Flint Water Crisis
The water crisis in Flint, Michigan, is an example of environmental injustice that took place beginning in 2014. The city decided to switch its drinking water supply from Detroit’s system to the water from the Flint River to save money. This switch was made despite inadequate treatment and testing of the water from the river. As a result, the community was deeply impacted. Their water supply turned yellow and began to smell and taste of sewage. Members of the community suffered from skin rashes and hair loss. Later, independent research would reveal that the contaminated water was contributing to dangerous increases of blood lead levels in the city’s youth, and the lead level in the water greatly exceeded the EPAs standard.
Eventually, in 2016, with the efforts of community activists and scientists, a federal judge sided with Flint residents and ordered clean bottled water to be delivered to citizens. The following year, the city was ordered to replace the city’s lead pipes with funding from the state and the community was provided with resources to support their health and well-being. However, the fight to bring safe access to the water supply is ongoing.
Read more about the water crisis in Flint, Michigan: https://www.nrdc.org/stories/flint-water-crisis-everything-you-need-know
Use Statistics to Defend Environmental Justice
To test whether the water was safe, the scientists likely conducted hypothesis tests. Now you will conduct a hypothesis test, like those scientists who fought for the citizens in Flint.
Step 1 Determine the hypotheses
Let \(\mu\) represent the _____________ lead level for the entire water supply.
\(H_0:\mu=\underline{\ \ \ \ \ \ \ \ \ \ }\text{ ppb}\)
Select the appropriate alternative hypothesis:
- \(H_a: \mu<15 \text{ ppb}\)
- \(H_a: \mu>15 \text{ ppb}\)
- \(H_a: \mu \neq 15 \text{ ppb}\)
Select the appropriate test, and explain why:
- Left-tailed
- Right-tailed
Step 2 Collect sample data
The sample mean, \(\bar{x}\), is __________ ppb.
The sample standard deviation, s, is __________ ppb.
The sample size, n, is _________ households.
There are __________ degrees of freedom.
Explain why the conditions of the Central Limit Theorem are met:
Step 3 Assess the evidence
Which distribution will we use to find the P-value?
- The normal distribution because the population standard deviation is given.
- The student’s T-distribution because the population standard deviation is unknown.
The test statistic, rounded to three decimal places, is
\[T=\frac{\bar{x}-\mu}{\frac{s}{\sqrt{n}}}=\frac{}{\frac{}{}}=\underline{\ \ \ \ \ \ \ \ \ \ }\nonumber\]
Below is the graph of the T-distribution with _______ degrees of freedom. Label the T-statistic on the horizontal axis. Then shade the region that represents the P-value.
Use https://www.desmos.com/calculator to find the P-value:
- Enter tdist(199) in the first line.
- Check the box for Find Cumulative Probability (CDF)
- The minimum and maximum will default to \(-\infty\) and \(\infty\) respectively. Enter the T-score in the min or max so that the graph matches the graph above.
Step 4 State a conclusion in context
The level of significance is \(\alpha=\underline{\ \ \ \ \ \ \ \ \ \ }\). The P-value (rounded to three decimal places) is 0.004. Fill in the blank with \(\leq\) or \(>\):
\[0.004\ \underline{\ \ \ \ \ \ \ \ \ \ }\ 0.01\nonumber\]
Defend the citizens of Flint:
The evidence supports the claim that the true _____________ lead level for the entire water supply in Flint is ___________________ _______________________ 15 ppb. Therefore, by the current standards, the lead level in Flint’s water supply is dangerously high.
Summary of Hypothesis Testing Process for a Single Population Mean
Step 1: determine the hypotheses.
In order to test a claim about a population parameter, we create two opposing hypotheses. We call these the null hypothesis, \(H_0\), and the alternative hypothesis, \(H_a\). Let \(\mu\) represent a given population mean.
The Null Hypothesis
In every hypothesis test, we assume that the null hypothesis is true. The null hypothesis is always a statement of equality and therefore, should always contain an equal symbol (=). When a test involves a single population mean, the null hypothesis will be
\[H_0: \mu=value\nonumber\]
The Alternative Hypothesis
The alternative hypothesis is a claim implied by the research question and is an inequality. The alternative hypothesis states that population mean is greater than (>), less than (<), or not equal (≠) to the assumed value in the null hypothesis.
When a test involves a single population mean, alternative hypothesis will be one of the following:
\[\begin{aligned} & H_a: \mu>\text { value } \\ & H_a: \mu<\text { value } \\ & H_a: \mu \neq \text { value } \end{aligned}\]
Step 2: Collect Sample Data
During a hypothesis test, we work to know if a sample statistic is unusual or not. Therefore, we must think about probabilities from a sampling distribution.
In a previous lesson, we learned about the sampling distribution of sample means. The Central Limit Theorem says that a sampling distribution of sample means is approximately normal if either the sample size, n, is greater than 30 or sampling was performed from a normally distributed population. In the second step of a hypothesis test, we verify that the sampling distribution is approximately normal and we identify or compute any sample statistics.
Step 3: Assess the Evidence
This step is all about probability. Since the sampling distribution is approximately normal (as determined in step 2), and the population standard deviation is likely unknown, we can compute a T-score and use the student’s T-distribution to find probabilities. The sampling distribution of sample means has mean
\[\mu_\bar{x}=\mu\nonumber\]
and standard error
\[\sigma_{\bar{x}}=\frac{s}{\sqrt{n}}\nonumber\]
where \(\mu\) is the assumed population mean, s is the sample standard deviation, and n is the sample size. The test statistic is
\[T=\frac{x-\mu}{\sigma} \text { which translates to } T=\frac{\bar{x}-\mu_{\bar{x}}}{\sigma_{\bar{x}}}=\frac{\bar{x}-\mu}{\frac{s}{\sqrt{n}}}\nonumber\]
when looking at the sampling distribution of sample means.
Step 4: State a Conclusion
Hypothesis tests are all about making decisions. We use the P-value to make a decision about the null and alternative hypotheses.
We compare our P-value to a level of significance. The level of significance, denoted \(\alpha\) (the greek letter “alpha”), is how unlikely a sample statistic needs to be to convince us about a claim. It is also the level of risk we accept in being wrong.
We have only two possible conclusions:
- If the P-value \(\leq \alpha\), we reject the null hypothesis and support the alternative hypothesis.
- This does not make the null hypothesis true—we cannot prove the null hypothesis because sample data cannot reveal the true value of the population mean.
| 15 | 15.4 | 16.2 | 16.1 | 15.8 | 16.2 |
| 15.7 | 15.6 | 16 | 16.3 | 15.3 | 15.9 |
Let \(\mu\)
represent _____________________________________________________________________
_____________________________________________________________________________
Which type of test and why?
The sampling distribution is approximately normal because ______________________________
______________________________________________________________________________
Go to https://www.desmos.com/calculator . Type C= into the first line and copy and paste the data.
\(\bar{x}=\operatorname{mean}(C)=\underline{\ \ \ \ \ \ \ \ \ \ }\text { ounces }\)
\(s=\operatorname{stdev}(C)=\underline{\ \ \ \ \ \ \ \ \ \ }\text { ounces }\)
\(n=\underline{\ \ \ \ \ \ \ \ \ \ }\text{ and }df=\underline{\ \ \ \ \ \ \ \ \ \ }\)
P-value is _________________________
Compare the P-value and the significance level. Make a decision, and state a conclusion in context:
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Next, the test statistic is used to conduct the test using either the p-value approach or critical value approach. The particular steps taken in each approach largely depend on the form of the hypothesis test: lower tail, upper tail or two-tailed. The form can easily be identified by looking at the alternative hypothesis (H a). If there is a ...
Decide on the alternative hypothesis: Use a two-tailed t-test if you only care whether the population's mean (or, in the case of two populations, the difference between the populations' means) agrees or disagrees with the pre-set value. ... Right-tailed. This t-test calculator allows you to use either the p-value approach or the critical ...
If this is not the case, you should instead use the Welch's t-test calculator. To perform a two sample t-test, simply fill in the information below and then click the "Calculate" button. Enter raw data Enter summary data. Sample 1. 301, 298, 295, 297, 304, 305, 309, 298, 291, 299, 293, 304. Sample 2.
Two tailed test example: A factory uses two identical machines to produce plastic plates. You would expect both machines to produce the same number of plates per minute. Let μ1 = average number of plates produced by machine1 per minute. Let μ2 = average number of plates produced by machine2 per minute. We would expect μ1 to be equal to μ2.
Use this Hypothesis Test Calculator for quick results in Python and R. Learn the step-by-step hypothesis test process and why hypothesis testing is important. Hypothesis Test Calculator | 365 Data Science
Enter the values for your two treatment conditions into the text boxes below, either one score per line or as a comma delimited list. Select your significance level and whether your hypothesis is one or two-tailed. Then give your data a final check, and press the "Calculate T and P Values" button. Treatment 1 ( X) Treatment 2 ( X) Significance ...
Two sample and one sample t-test calculator with step by step explanation. Site map; Math Tests; Math Lessons; Math Formulas; Calculators; ... Two Tailed Test (default) One Tailed Test: 3. Significance Level: 0.05 (default) 0.01: 0.001: 4. Choose a test Unpaired T Test (default)
A t test compares the means of two groups. There are several types of two sample t tests and this calculator focuses on the three most common: unpaired, welch's, and paired t tests. Directions for using the calculator are listed below, along with more information about two sample t tests and help on which is appropriate for your analysis. NOTE: This is not the same as a one sample t test; for ...
Finding p value from the Pearson (r) score involves the following steps: Calculate the test statistic (t) t= r n-2 (1 - r2) Determine the degrees of freedom (df) = n−2. Use the t-distribution table to determine the critical t-value and interpolate (if necessary) y = y + (x - x 1 ) (y 2 - y 1) x 2 - x 1. One Tail.
T-Test Calculator for 2 Independent Means. This simple t -test calculator, provides full details of the t-test calculation, including sample mean, sum of squares and standard deviation. A t -test is used when you're looking at a numerical variable - for example, height - and then comparing the averages of two separate populations or groups (e.g ...
Two-tailed hypothesis tests are also known as nondirectional and two-sided tests because you can test for effects in both directions. When you perform a two-tailed test, you split the significance level percentage between both tails of the distribution. ... Use SPSS for two-tailed tests to calculate the t & p values. Then report the p-value as ...
Quick P Value from Z Score Calculator. P Value from Z Score Calculator. This is very easy: just stick your Z score in the box marked Z score, select your significance level and whether you're testing a one or two-tailed hypothesis (if you're not sure, go with the defaults), then press the button! If you need to derive a Z score from raw data ...
Choose the alternative hypothesis: two-tailed or left/right-tailed. In our Z-test calculator, you can decide whether to use the p-value or critical regions approach. In the latter case, set the significance level, α. \alpha α. Enter the value of the test statistic, z. z z.
The p-value is for a one-sided hypothesis (one-tailed test), allowing you to infer the direction of the effect (more on one vs. two-tailed tests). However, the probability value for the two-sided hypothesis (two-tailed p-value) is also calculated and displayed, although it should see little to no practical applications.
Instructions: Use this calculator to work on a two-samples t-test, showing all the steps. In order to run the test, you need two provide two independent samples in the spreadsheet below. You can either type the data or simply paste them from Excel. Ho: \mu_1 μ1 \mu_2 μ2. Ha: \mu_1 μ1 \mu_2 μ2. Significance Level ( \alpha α) =.
For example, in a two-tailed Z test with critical values -1.96 and 1.96 (corresponding to 0.05 significance level) the critical regions are from -∞ to -1.96 and from 1.96 to +∞. Therefore, if the statistic falls below -1.96 or above 1.96, the null hypothesis test is statistically significant.
Plugging these values into the One Sample t-test Calculator, we obtain the following results: t-test statistic: 2.1689; two-tailed p-value: 0.0478; ... This is an example of a two-tailed hypothesis test because the alternative hypothesis contains the not equal "≠" sign. The professor believes that the studying technique will influence the ...
A two-sample t-test always uses the following null hypothesis: H 0: μ 1 = μ 2 (the two population means are equal) The alternative hypothesis can be either two-tailed, left-tailed, or right-tailed: H 1 (two-tailed): μ 1 ≠ μ 2 (the two population means are not equal) H 1 (left-tailed): μ 1 < μ 2 (population 1 mean is less than population ...
Welcome to the critical value calculator! Here you can quickly determine the critical value (s) for two-tailed tests, as well as for one-tailed tests. It works for most common distributions in statistical testing: the standard normal distribution N (0,1) (that is when you have a Z-score), t-Student, chi-square, and F-distribution.
With Python use the scipy and math libraries to calculate the P-value for a two tailed hypothesis test for a mean. Here, the sample size is 30, the sample mean is 62.1, the sample standard deviation is 13.46, and the test is for a mean different from 60. ... With R use built-in math and statistics functions find the P-value for a two tailed ...
The one sample t-test, also called single sample t-test, compares the sample mean to the population mean. more How to use the one sample t test calculator? Choose the tails: The one-tailed test, left or right, is more powerful than the two-tailed test and results in a smaller p-value, half since the t distribution is symmetrical.
This calculator will conduct a complete one-sample t-test, given the sample mean, the sample size, the hypothesized mean, and the sample standard deviation. The results generated by the calculator include the t-statistic, the degrees of freedom, the critical t-values for both one-tailed (directional) and two-tailed (non-directional) hypotheses, and the one-tailed and two-tailed probability ...
It is the alternative hypothesis that determines what "extreme" actually means, so the p-value depends on the alternative hypothesis that you state: left-tailed, right-tailed, or two-tailed. In the formulas below, S stands for a test statistic, x for the value it produced for a given sample, and Pr(event | H 0 ) is the probability of an event ...
The alternative hypothesis is a claim implied by the research question and is an inequality. The alternative hypothesis states that population mean is greater than (>), less than (<), or not equal (≠) to the assumed value in the null hypothesis. When a test involves a single population mean, alternative hypothesis will be one of the following: