5 Each person in a random sample of 15 men and 17 women from a university campus was asked how many days in a month they took exercise. The numbers of days for men and women, \(x _ { M }\) and \(x _ { W }\) respectively, are summarised by $$\Sigma x _ { M } = 221 , \quad \Sigma x _ { M } ^ { 2 } = 3992 , \quad \Sigma x _ { W } = 276 , \quad \Sigma x _ { W } ^ { 2 } = 5538 .$$

1. State conditions for the validity of a suitable test of the difference in the mean numbers of days for men and women on the campus.
2. Given that these conditions hold, carry out the test at the \(5 \%\) significance level.
3. If in fact the random sample was drawn entirely from the university Mathematics Department, state with a reason whether the validity of the test is in doubt.