t Test for Two Related Samples (Repeated Measures)

Repeated measures? Whenever the same subject is measured more than once. Two related samples occur whenever each observation in one sample is paired, on a one-to-one basis, with a single observation in the other sample.

What is compared? The mean difference scores between the two groups. D = Σ D n The sign of D is crucial.

Problems with repeated measures: Enough time must pass between measures to ensure no bias or lingering effects. Counterbalancing – half of the subjects experience the conditions in the opposite order. A then B or B then A.

Hypotheses Null Hypothesis H 0 : μ D = 0 Alternative Hypothesis Directional H 1 : μ D > 0 or H 1 : μ D < 0 Non Directional H 1 : μ D ≠ 0

t ratio for two population means (two related samples) t = sample mean difference – hypothesized population mean difference estimated standard error or D - µ Dhyp s D

Calculations 1. Assign a value to n, the number of difference scores 2. Subtract X2 from X1 to obtain D 3. Sum all D scores 4. Calculate mean of D 5. Calculate SS for D 6. Find standard error S D 7. Solve for t

Use the EPO data Scores for Two EPO Experiments X1X2D

Use the EPO data (p 323) Scores for Two EPO Experiments X1X2DD2D

Calculations SS D = ΣD 2 – S D = (ΣD) 2 n √ SS D n - 1 SDSD √ n

Calculations t = D – µ D hyp S D

Confidence interval (p 319) D ± (t conf )(s D ) Find value of t conf in Table B

Standardized Effect Size, Cohen’s d d = D s D

Progress Check 15.2 Days Ill Due to Colds Pair NumberVitamin C (X 1 )Fake Vitamin C (X 2 )

t test for population correlation (p329) t = ρ hyp = 0 r - ρ hyp √ 1 – r 2 n - 2

Progress Check 15.6 (p 331) A random sample of 27 California taxpayers reveals an r =.43 between years of education and annual income. Use t to test the null hypothesis at the.05 level of significance that there is no relationship between educational level and annual income for the population of California taxpayers. Answer on 511.