An employee of a university's recreational facility is interested in the effect of exercise in maintaining mental ability. She decides to study the faculty members aged 40 to 50 at her university, looking separately at two groups: the ones that exercise regularly, and the ones that don't. There are many people in each group, so she takes a simple random sample of 120 people from each group for detailed study, with the samples taken independently of each other.

One of the things she does is to give an IQ test to everyone in the study. Those who exercise regularly have an average IQ of 135, with an SD of 15, and those who don't exercise regularly have an average IQ of 125, with an SD of 20.

(a) Carry out all the steps of a two-sample z test to evaluate the null hypothesis that these two populations have the same average IQ.

(b) Suppose we wanted to use the results of this study to conclude that exercise does indeed help to maintain mental ability. What is the problem with this?