# 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:

- Calculate the difference between pre and post values, call it \(d\)
- Calculate the mean of this difference
- Calculate the standard deviation of this difference
- Use the standard deviation to calculate the standard error.
- Use the mean and standard deviation in the Paired Sample T-Test formula.

Calculations:

- is simply: \(d = post - pre\)

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

\(d\) | f | fd | \(d - \overline{d}\) | \((d - \overline{d})^2\) | \(f(d - \overline{d})^2\) |
---|---|---|---|---|---|

… | … | … | … | … | … |

. | \(\sum f\) | \(\sum fd\) | \(\sum = SS_d\) |

- Calculate the mean of this difference

\[\overline{d} = \frac{\sum fd}{\sum f}\]

- Calculate the standard deviation of this difference

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

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

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.

- What is your null hypothesis?
- What is the alternative hypothesis?
- What is your alpha?
- Interpret your p-value.