**Instructions:** This calculator conducts a t-test for two paired samples. This test applies when you have two samples that are dependent (paired or matched). Please select the null and alternative hypotheses, type the sample data and the significance level, and the results of the t-test for two dependent samples will be displayed for you:

#### The T-Test For Paired Samples

More about the *t-test for two dependent samples* so you can understand in a better way the results delivered by the solver: A t-test for two paired samples is a hypothesis test that attempts to make a claim about the population means (\(\mu_1\) and \(\mu_2\)). More specifically, a t-test uses sample information to assess how plausible it is for difference \(\mu_1\) - \(\mu_2\) to be equal to zero. The test has two non-overlaping hypotheses, the null and the alternative hypothesis. The null hypothesis is a statement about the population parameter which indicates no effect, and the alternative hypothesis is the complementary hypothesis to the null hypothesis. The main properties of the t-test for two paired samples are:

- The test required two dependent samples, which are actually paired or matched or we are dealing with repeated measures (measures taken from the same subjects)
- As with all hypotheses tests, depending on our knowledge about the "no effect" situation, the t-test can be two-tailed, left-tailed or right-tailed
- The main principle of hypothesis testing is that the null hypothesis is rejected if the test statistic obtained is sufficiently unlikely under the assumption that the null hypothesis is true
- The p-value is the probability of obtaining sample results as extreme or more extreme than the sample results obtained, under the assumption that the null hypothesis is true
- In a hypothesis tests there are two types of errors. Type I error occurs when we reject a true null hypothesis, and the Type II error occurs when we fail to reject a false null hypothesis

The formula for a t-statistic for two dependent samples is:

\[t = \frac{\bar D}{s_D/\sqrt{n}}\]where \(\bar D = \bar X_1 - \bar X_2\) is the mean difference and \(s_D\) is the sample standard deviation of the differences \(\bar D = X_1^i - X_2^i\), for \(i=1, 2, ... , n\).

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