D.27 Answers: Tests for paired means

Answers to exercises in Sect. 29.11.

Answer to Exercise 29.1: \(H_0\): \(\mu_d = 0\) and \(H_1\): \(\mu_d>0\): differences are positive when the dip rating is better than the raw rating. \(t = (5.2 - 0)/3.06 = 1.699\); \(P\) larger than 0.05: the evidence 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. \(H_0\): \(\mu_d = 0\) and \(H_1\): \(\mu_d>0\) (because the differences are reductions). 5. \(t=8.12\). 6. \(P = 0.001\div 2 = 0.0005\) (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: \(H_0\): \(\mu_d = 0\) and \(H_1\): \(\mu_d>0\), where differences are positive when the intention to smoke is reduced after exercise. \(t = (0.66 - 0)/0.37 = 1.78\); \(P\)-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: \(H_0\): \(\mu_d = 0\) and \(H_1\): \(\mu_d>0\), where differences refer to the reduction in ferritin. \(\bar{d} = -424.25\) 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