## D.23 Answers: CIs for two means

Answers to exercises in Sect. 24.13.

Answer to Exercise 24.1: 1. Table D.3. 2. From SPSS, exact 95% CI: 0.05438 to 0.11501 (bottom row). Exact 95% CI for the difference between the mean direct HDL cholesterol concentrations: 0.05438 to 0.11501 mm Hg higher for non-smokers.
TABLE D.3: A summary table for the NHANES data; statistics in mm Hg
Mean Standard deviation Standard error Sample size
Non-smokers 1.3924 0.42792 0.01048 1668
Smokers 1.3077 0.42353 0.01137 1388
Differences 0.0847 0.01546

Answer to Exercise 24.2: 1. Placebo group: $$3.62/\sqrt{176} = 0.2728678$$ days; echinacea group $$3.31/\sqrt{183} = 0.2446822$$ days. 2. $$0.53 \pm (2\times 0.367)$$, or -0.204 to 1.264 days. 3. Placebo minus echinacea: the difference between the means show how much longer symptoms last with placebo, compared to echinacea. 4. $$6.34 \pm (2\times 0.2446822)$$, or 5.85 to 6.83 days. 5. Sample sizes are large, so the CIs statistically valid.

The difference between the means is an average of 0.53 days; about half a day (quicker on echinacea). Probably not that important when a cold last for almost seven days.
Answer to Exercise 24.3: 1. Exercise group: $$1.4/\sqrt{10} = 0.4427189$$; splinting group: $$1.1/\sqrt{10} = 0.3478505$$. 2. Splinting minus exercise: the difference are how much greater the pain is with splinting. 3. $$0.3\pm (2\times 0.563$$, or from -0.826 to 1.426: 0.826 greater pain with exercise to 1.426 greater pain with splinting. 4. $$1.1\pm (2\times 0.3478505)$$: from 0.404 to 1.796. 5. Sample sizes are small; CIs may not be statistically valid, roughly correct only.