Consider the following data from a repeated-measures design. You want to use a repeated-measures t-test to test the null hypothesis H₀: μD = 0 (the null hypothesis states that the mean difference for the general population is zero). The data consist of five observations, each with two measurements, A and B, taken before and after a treatment. Assume the population of the differences in these measurements are normally distributed. Complete the following table by calculating the differences and the squared differences:
Observation A B Difference Score Squared Difference Score (D = B – A) (D²)
1 1 2 1 1
2 2 3 1 1
3 3 4 1 1
4 6 4 -2 4
5 5 6 1 1
6 8 5 -3 9
The mean difference score is MD = -0.5.
For a repeated-measures t-test, you need to calculate the t statistic, which requires you to calculate s and sMD.
What is the estimated standard deviation of the difference scores?
s = √(ΣD² / n - 1) = √(17 / 5) = 1.30
What is the estimated standard error of the mean difference scores? (Note: For best results, retain at least six decimal places from your calculation of s.)
sMD = s / √n = 1.30 / √6 = 0.581
What is the t statistic for the repeated-measures t-test to test the null hypothesis H₀: μD = 0?
t = MD / sMD = -0.5 / 0.581 = -0.861
t Distribution
Degrees of Freedom = 5
You conduct a two-tailed test at α = .05. To use the Distributions tool to find the critical values, you first need to set the degrees of freedom in the tool. The degrees of freedom are 5.
The critical values (the values for t scores that separate the tails from the main body of the distribution, forming the critical region) are -2.571 and 2.571.
Finally, since the t statistic -0.861 is not in the critical region, you fail to reject the null hypothesis.