*57*

A paired samples t-test is used to compare the means of two samples when each observation in one sample can be paired with an observation in the other sample.

The following step-by-step example shows how to perform a paired samples t-test to determine if the population means are equal between the following two groups:

**Step 1: Calculate the Test Statistic**

The test statistic of a paired t-test is calculated as:

**t = x _{diff}Â / (s_{diff}/âˆšn)**

where:

**x**sample mean of the differences_{diff}:Â**s:Â**sample standard deviation of the differences**n:Â**sample size (i.e. number of pairs)

We will calculate the mean of the differences between the two groups and the standard deviation of the differences between the two groups:

Thus, our test statistic can be calculated as:

- t = x
_{diff}Â / (s_{diff}/âˆšn) - t = 1.75 / (1.422/âˆš12)
- t =Â
**4.26**

**Step 2: Calculate the Critical Value**

Next, we need to find the critical value to compare our test statistic to.

For this example, weâ€™ll use a two-tailed test with Î± = .05 and df = n-1 degrees of freedom.

According to the t-Distribution table, the critical value that corresponds to these values is **2.201**:

**Step 3: Reject or Fail to Reject the Null Hypothesis**

Our paired samples t-test uses the following null and alternative hypothesis:

**H**Î¼_{0}:_{1}= Î¼_{2}(the two population means are equal)**H**Î¼_{A}:â‰ Î¼_{1}_{2}(the two population means are not equal)

Since the absolute value of our test statistic (**4.26**) is greater than the critical value found in the t-table (**2.201**), we reject the null hypothesis.

This means we have sufficient evidence to say that the mean between the two groups is not equal.

**Bonus:** Feel free to use the Paired Samples t-test Calculator to confirm your results.