A health magazine reported that 27% of adult males smoke cigarettes. A competing magazine thinks this claim is false.

Question

A health magazine reported that 27% of adult males smoke cigarettes. A competing magazine thinks this claim is false. To test this claim, they select 40 adult males at random and record if they smoke cigarettes. They calculate a 95% confidence interval for the proportion of adult males who smoke and find it is 0.22 to 0.28. 

Interpret the confidence interval

Answer 1

This proposal is invalid. It is not possible to have a 95% confidence interval with these bounds when the sample size is only 40, IF you are basing it on a normal distribution.

A binomial distribution is approximately normal ONLY if np(1-p) ≥ 10.

For this example n = 40 and p = 0.25 (the average of the endpoints of the interval, i.e. the midpoint)

Thus

40 x 0.25 x (1 - 0.25) = 7.5 < 10.

This is a skewed distribution which cannot use a normal confidence interval.