Paired t-Test Math Example 1

Follow the full solution, then compare it with the other examples linked below.

Example 1

medium
Students' scores before and after tutoring: Before: {70,65,80,75,60}\{70, 65, 80, 75, 60\}, After: {75,70,85,80,70}\{75, 70, 85, 80, 70\}. Conduct a paired t-test at ฮฑ=0.05\alpha=0.05 to test if tutoring improved scores.

Solution

  1. 1
    Differences di=Afterโˆ’Befored_i = \text{After} - \text{Before}: 5,5,5,5,105, 5, 5, 5, 10
  2. 2
    dห‰=(5+5+5+5+10)/5=30/5=6\bar{d} = (5+5+5+5+10)/5 = 30/5 = 6; sd=โˆ‘(diโˆ’dห‰)2nโˆ’1=4+1+1+1+164=4.25โ‰ˆ2.06s_d = \sqrt{\frac{\sum(d_i-\bar{d})^2}{n-1}} = \sqrt{\frac{4+1+1+1+16}{4}} = \sqrt{4.25} \approx 2.06
  3. 3
    t-statistic: t=dห‰sd/n=62.06/5=60.921โ‰ˆ6.51t = \frac{\bar{d}}{s_d/\sqrt{n}} = \frac{6}{2.06/\sqrt{5}} = \frac{6}{0.921} \approx 6.51
  4. 4
    df=4; t0.05,4โˆ—=2.132t^*_{0.05, 4} = 2.132 (one-tailed); 6.51>2.1326.51 > 2.132 โ†’ Reject H0H_0

Answer

tโ‰ˆ6.51>2.132t \approx 6.51 > 2.132. Reject H0H_0. Tutoring significantly improved scores.
A paired t-test analyzes differences between matched pairs (before-after, or twins). By working with differences, we eliminate between-subject variability, increasing power. The test becomes a one-sample t-test on the differences.

About Paired t-Test

A hypothesis test for the mean difference in a paired (matched) data design, where each subject provides two related measurements. The test analyzes the differences di=x1iโˆ’x2id_i = x_{1i} - x_{2i} as a single sample.

Learn more about Paired t-Test โ†’

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