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=(5.2−0)/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.

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=(0.66−0)/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:
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.

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