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.

HypothesesNull Hypothesis

H0: μD = 0

Alternative HypothesisDirectional

H1: μD > 0 or H1: μD < 0

Non Directional H1: μD ≠ 0

t ratio for two population means(two related samples)

t = sample mean difference – hypothesized population mean difference

estimated standard error

or D - µDhyp

sD

Calculations1. Assign a value to n, the number of

difference scores2. Subtract X2 from X1 to obtain D3. Sum all D scores4. Calculate mean of D5. Calculate SS for D6. Find standard error SD

7. Solve for t

Use the EPO dataScores for Two EPO Experiments

X1 X2 D

12 7 5

5 3 2

11 4 7

11 6 5

9 3 6

18 13 5

Use the EPO data (p 323)

Scores for Two EPO Experiments

X1 X2 D D2

12 7 5 25

5 3 2 4

11 4 7 49

11 6 5 5

9 3 6 36

18 13 5 25

30 164

Calculations

SSD = ΣD2 –

SD =

SD =

(ΣD) 2

n

√SSD

n - 1

SD

√ n

Calculations

t = D – µDhyp

SD

Confidence interval (p 319)

D ± (tconf)(sD)

Find value of tconf in Table B

Standardized Effect Size, Cohen’s d

d = D sD

Progress Check 15.2Days Ill Due to Colds

Pair Number Vitamin C (X1) Fake Vitamin C (X2)

1 2 3

2 5 4

3 7 9

4 0 3

5 3 5

6 7 7

7 4 6

8 5 8

9 1 2

10 3 5

t test for population correlation (p329)

t =

ρhyp = 0

r - ρhyp

√1 – r2

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.