Paired t-Test Walkthrough Cheat Sheet

A paired t-test is used when two quantitative measurements are linked, such as before-and-after scores from the same students or measurements from matched pairs. This cheat sheet helps students organize the test from raw paired data to a final conclusion. It focuses on creating differences, checking conditions, computing the test statistic, and interpreting the result in context.

The paired t-test is important because it analyzes change within pairs instead of treating the two samples as unrelated.

The key idea is to subtract within each pair and then run a one-sample t-test on the list of differences. The main formulas are the sample mean difference $\bar{d}$ , the sample standard deviation of differences $s_d$ , the standard error $SE = \frac{s_d}{\sqrt{n}}$ , and the test statistic $t = \frac{\bar{d} - \mu_{d,0}}{s_d / \sqrt{n}}$ . The degrees of freedom are $df = n - 1$ , where $n$ is the number of pairs.

A confidence interval for the mean difference is $\bar{d} \pm t^{*}\frac{s_d}{\sqrt{n}}$ .

Key Facts

A paired t-test uses differences $d_i = x_{i,2} - x_{i,1}$ from matched measurements or the same subject measured twice.
The null hypothesis is usually $H_0: \mu_d = 0$ , meaning the true mean difference is zero.
The alternative hypothesis can be $H_a: \mu_d > 0$ , $H_a: \mu_d < 0$ , or $H_a: \mu_d \ne 0$ depending on the question.
The sample mean difference is $\bar{d} = \frac{\sum d_i}{n}$ , where $n$ is the number of paired differences.
The standard error of the mean difference is $SE = \frac{s_d}{\sqrt{n}}$ , where $s_d$ is the sample standard deviation of the differences.
The paired t-test statistic is $t = \frac{\bar{d} - \mu_{d,0}}{s_d / \sqrt{n}}$ , where $\mu_{d,0}$ is the hypothesized mean difference.
The degrees of freedom for a paired t-test are $df = n - 1$ .
A confidence interval for the true mean difference is $\bar{d} \pm t^{*}\frac{s_d}{\sqrt{n}}$ .

Vocabulary

Paired data: Paired data are two related measurements that belong together, such as pretest and posttest scores for the same person.
Difference: A difference is the value found by subtracting one measurement in a pair from the other, written as $d_i = x_{i,2} - x_{i,1}$ .
Mean difference: The mean difference $\bar{d}$ is the average of all paired differences in the sample.
Standard error: The standard error $SE = \frac{s_d}{\sqrt{n}}$ estimates how much the sample mean difference varies from sample to sample.
Degrees of freedom: Degrees of freedom for a paired t-test are $df = n - 1$ , based on the number of paired differences.
P-value: A p-value is the probability of getting a test statistic as extreme as the observed one if the null hypothesis is true.

Common Mistakes to Avoid

Treating paired data as independent samples is wrong because the two measurements in each pair are related. Use the differences $d_i$ and perform a one-sample t-test on those differences.
Subtracting in an inconsistent order is wrong because it can reverse the sign of $\bar{d}$ and change the interpretation. Choose one order, such as $d_i = \text{after} - \text{before}$ , and use it throughout.
Using $n$ as the number of individual measurements is wrong because $n$ is the number of pairs. If there are $12$ people measured twice, then $n = 12$ , not $n = 24$ .
Using $df = n$ is wrong because one degree of freedom is lost when estimating $\bar{d}$ . For a paired t-test, always use $df = n - 1$ .
Interpreting the result without context is incomplete because the test is about the mean difference, not just a generic mean. State the conclusion using the variable, direction of subtraction, and population.

Practice Questions

1 A teacher records quiz scores before and after a study strategy for $10$ students. The differences are defined as $d_i = \text{after} - \text{before}$ , with $\bar{d} = 4.2$ and $s_d = 3.0$ . Test $H_0: \mu_d = 0$ using the test statistic formula and find $t$ .
2 For a paired t-test with $n = 16$ pairs, $\bar{d} = -2.5$ , $s_d = 5.0$ , and $\mu_{d,0} = 0$ , compute $SE = \frac{s_d}{\sqrt{n}}$ and $t = \frac{\bar{d} - \mu_{d,0}}{SE}$ .
3 A sample of $25$ matched pairs has $\bar{d} = 1.8$ and $s_d = 4.5$ . Find the degrees of freedom and write the form of a $95\%$ confidence interval as $\bar{d} \pm t^{*}\frac{s_d}{\sqrt{n}}$ .
4 A researcher measures blood pressure before and after a new exercise program for the same people. Explain why a paired t-test is more appropriate than a two-sample t-test.

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Paired t-Test Walkthrough Cheat Sheet

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