D.27 Answers: Tests for paired means

Answers to exercises in Sect. 29.12.

Answer to Exercise 29.1: H0: μd=0 and H1: μd>0: differences are positive when the dip rating is better than the raw rating. t=(5.20)/3.06=1.699; the approximate one-tailed P-value, from using the 68–95–99.7 rule, is somewhere between 16% and 2.5%. So we cannot be sure if the P-value is larger than 0.05… but it is likely that it is (since the calculated t-score is quite a long distance from z=1). the evidence probably doesn’t support the alternative hypothesis.
Answer to Exercise 29.2: 1. Because it is the bloood pressure reduction, and a reduction is what the drug is meant to produce, so expect the reductions to be positive nunmbers. 2. Differences shown below. 3. Histogram of differences: Fig. D.7. 4. H0: μd=0 and H1: μd>0 (because the differences are reductions). 5. t=8.12. 6. P=0.001÷2=0.0005 (one-tailed test). 7. Very strong evidence (P=0.0005) that the drug reduces the average systolic blood pressure (mean reduction: 8.6 mm Hg) in the population.
A histogram of the systolic blood pressure reductions (in mm Hg)

FIGURE D.7: A histogram of the systolic blood pressure reductions (in mm Hg)

Answer to Exercise 29.3: H0: μd=0 and H1: μd>0, where differences are positive when the intention to smoke is reduced after exercise. t=(0.660)/0.37=1.78; P-value larger than 0.05: the evidence doesn’t support the alternative hypothesis. No evidence (P>0.05) that the mean ‘intention to smoke’ reduced after exercise in women (mean change in intention to smoke: -0.66; std. error: 0.37).
Answer to Exercise 29.4: H0: μd=0 and H1: μd>0, where differences refer to the reduction in ferritin. ˉd=424.25 and s=2092.693 and n=20, so t=0.90663. t is ‘small’; P>0.05 (actually P=0.376): the evidence doesn’t support the alternative hypothesis. Since n<25, the test may not be statistically valid (the histogram of data (Fig. D.8) suggests that the population might have a normal distribution), though the P-value is very large so it probably makes little difference.
A histogram of the change in ferritin concentration

FIGURE D.8: A histogram of the change in ferritin concentration