1. COMPARING TWO POPULATIONS IDEA: Compare two groups/populations based on samples from each of them. Examples. Compare average height of men and women. Draw sample of men heights: x1, x2, …, xm and a sample of women heights: y1, y2, …, yn. Test Ho: Ho: μx = μy vs Ha: μx ≠ μy or Ha: μx > μy Compare proportions of Democrats in two cities, Compare weights of people before and after a diet, etc. General considerations for the samples: Dependent or independent samples. Example. Comparing weights of people before and after a diet we have dependent (same people) samples of weights. Comparing weights of people in two cities we have independent samples. Analysis methods will differ for dependent and independent samples.

2. PAIRED t-TEST: dependent samples Observations come as matched pairs (X,Y). X and Y are NOT independent, X and Y are dependent. Examples. X is score on a test before studying hard; Y is score on the test after studying hard for the same student; X is score on a test or in sports before training program, Y score after training program; X is weight before weight loss program, Y is weight after the program; X and Y are heights of twins or siblings.

3. PAIRED t-TEST: HYPOTHESES Hypotheses of interest: does training make a difference? μx = score before training; μy = score after training. Ho: μx = μy vs Ha: μx < μy (no difference) (score after training is higher) Data are pairs of observations: (x1, y1), (x2, y2), …, (xn, yn). Typically, we work with differences: d=X-Y, and phrase hypotheses in terms of differences: μd = true mean difference. obsbeforeafterdifference 1x1y1d1=x1-y1 2x2y2d2=x2=y2.... nxnyndn=xn-yn In terms of differences: Hypotheses e.g. Ho: μd = 0 vs Ha: μd < 0 Data: d1, d2, …, dn.

4. PAIRED t-TEST: TEST PROCEDURE To test Ho, we do one sample t-test. Need sample mean and standard deviation of d’s: Compute the test statistic: Under Ho the test statistic has t(n-1) distribution. Make decision in exactly the same way as for the one sample t-test. A (1-α)100% CI for d:

5. PAIRED t-TEST: an example The amount of lactic acid in the blood was examined for 10 men, before and after a strenuous exercise, with the results in the following table. (a) Test if exercise changes the level of lactic acid in blood. Use significance level α=0.01. (b) Find a 95% CI for the mean change in the blood lactose level. Before151613 172013161418 After33203035403718262119

6. PAIRED t-TEST: lactic acid example contd. Solution. Take d=“After level” – “before level” of lactic acid. Data for d: 18, 4, 17, 22, 23, 17, 5, 10, 7, 1. Sample stats: STEP1. Ho: μd = 0 vs Ha: μd ≠ 0 STEP 2. Test statistic: STEP 3. Critical value? df=n-1=9, t α/2 =t 0.005 =3.69. STEP 4. DECISION: t = 4.93 > 3.69 = t 0.005, so reject Ho. STEP 5. Exercise changes lactic acid level.

7. Example contd. (b) Find a 95% CI for the mean change in the blood lactose level. It is the familiar formula for the 95% CI for the mean, this time mean difference μd. Need percentile from the t distribution with n-1 degrees of freedom. n=10, n-1=9, α=0.05, so t α/2 =t 0.025 =2.262, so the 95% CI for μd is:

8. Lactic acid example in MINITAB: data set lactic-acid.MPJ

9. Paired T-Test and Confidence Interval Paired T for before - after N Mean StDev SE Mean before 10 15.50 2.37 0.75 after 10 27.90 8.17 2.58 Difference 10 -12.40 7.95 2.51 95% CI for mean difference: (-18.08, -6.72) T-Test of mean difference = 0 (vs not = 0): T-Value = -4.93 P-Value = 0.001 Conclusion: Reject Ho, lactic acid level changes after exercise. Note: CI for “Before –after” HoHa