## D.22 Answers: CIs for paired data

Answers to exercises in Sect. 23.12.

Answer to Exercise 23.1: Mean of the differences: 5.2; standard error 3.6. Approximate 95% CI: $$5.2 \pm (2\times 3.06)$$, or $$5.2\pm 6.12$$, from -0.92 to 11.22. Mean taste preference between preferring it better with dip by up to 11.2mm on the 100mm visual analog scale, or preferring it without dip by a little (up to -0.9mm on the 100mm visual analog scale. (Understanding how the differences are defined is needed to understand where this came from.)

A useful summary might be like Table D.2.
TABLE D.2: A numerical summary for the brocilli data
Mean Standard deviation Standard error
Raw 56 26.6 2.64679892595857
With dip 61.2 28.7 2.85575673590267
Differences 5.2 3.06
Answer to Exercise 23.2: 1. Computing differences as Before minus the After measurements seems sensible: the average blood pressure decrease, the purpose of the drug. 2. The differences (when defined asreductions): 9, 4, 21, 3, 20, 31, 17, 26, and so on. 3. Mean difference: 18.933; standard deviation: 9.027; standard error: $$9.027/\sqrt{15} = 2.331$$. Approximate 95% CI: $$14.271$$ to $$23.56$$ mm Hg. 4. Exact 95% CI: 13.934 to 23.93 mm Hg from output. 5. The first uses approximate multipiers. The second uses exact multipliers.
Answer to Exercise 23.3: 1. Approximate 95% CI for reduction: $$0.66 \pm(2\times 0.37)$$, or -0.08 to 1.4: average could be an increase of up to 0.08 to a reduction of up to 1.4 on the given scale for women. 2. Sample size is not larger than 25, but close: probably reasonably statistically valid.