Topic 8 Paired Sample T-Test

Before, with the independent sample t-test, we wanted to test if the population mean of a group was equal to the population mean of another (independent) group.

Now, we want to do a similar analysis where groups are no longer independent. We want to compare the the means of a variable for the same individuals, at different times periods.

This is called the Paired Sample T-Test

We will use this test to examine the means of one continuous across different time periods.

8.1 Formulas and calculations

Steps:

1. Calculate the difference between pre and post values, call it $d$
2. Calculate the mean of this difference
3. Calculate the standard deviation of this difference
4. Use the standard deviation to calculate the standard error.
5. Use the mean and standard deviation in the Paired Sample T-Test formula.

Calculations:

1. is simply: $d = post - pre$

To perform (2) and (3), you need to use the following table:

$d$	f	fd	$d - \overline{d}$	$(d - \overline{d})^2$	$f(d - \overline{d})^2$
…	…	…	…	…	…
.	$\sum f$	$\sum fd$			$\sum = SS_d$

2. Calculate the mean of this difference

$$\overline{d} = \frac{\sum fd}{\sum f}$$

3. Calculate the standard deviation of this difference

$$sd = \sqrt{\frac{\sum SS_d}{n-1} }$$

4. Use the standard deviation to calculate the standard error. $$se = \frac{sd}{\sqrt{n}}$$

5. Use the mean and standard deviation in the Paired Sample T-Test formula.

$$t_{paired} = \frac{\overline{d}}{se}$$

8.2 Interpretation

Compare your calculated $t$ to the $t_{critical}$ from the t-table.

• if the calculated $t$ is higher than the critical value ($t_{critical}$), we reject the null hypothesis.
• if the calculated $t$ is lower than the critical value ($t_{critical}$), we do not reject the null hypothesis.

8.3 Interpretation of SPSS results

Once again, we look at the p-value:

• $p \leq \alpha$ we reject the null
• $p > \alpha$ we fail to reject the null

8.4 Exercise

First, I will illustrate with the sample test scores dataset.

Second, using the “school-data.sav” do the necessary procedures to check if the there was an increase in api scores between 1999 and 2000 in the schools in the sample.

1. What is your null hypothesis?
2. What is the alternative hypothesis?
3. What is your alpha?
4. Interpret your p-value.