Conditions for Paired t-Test

1. Random: Pairs are randomly selected or treatments randomly assigned within pairs
2. 10%: Number of pairs $< 10\%$ of all possible pairs
3. Normal: The differences are approximately Normal (check with graph of differences)

Paired t-Confidence Interval

$\bar{d} \pm t^* \frac{s_d}{\sqrt{n}}$

Paired vs. Two-Sample: How to Decide

Why Pairing Helps

Pairing reduces variability by controlling for individual differences. Each subject serves as their own control, so person-to-person variation is removed.

Example: Testing a new study method

• Paired design: Same students take two tests (before and after method)
• Two-sample design: Different students use different methods
• The paired design is more powerful because it eliminates student-to-student variability

Common Mistakes

1. Using a two-sample test when data is paired
2. Forgetting to check Normality of the differences (not the original data)
3. Computing differences inconsistently (always subtract in the same direction)

AP Tip: If the data has a natural pairing, you MUST use a paired t-test. Using a two-sample t-test on paired data is incorrect and will cost points.