1. As part of their research two sports science students, Ali and Bea, select a random sample of 10 adult male swimmers and a random sample of 13 adult male athletes from local sports clubs. They measure the arm span, \(x \mathrm {~cm}\), of each person selected.
   The data are summarised in the table below

The students know that the arm spans of adult male swimmers and of adult male athletes may each be assumed to be normally distributed.
They decide to share out the data analysis, with Ali investigating the means of the two distributions and Bea investigating the variances of the two distributions. Ali assumes that the variances of the two distributions are equal. She calculates the pooled estimate of variance, \(s _ { p } { } ^ { 2 }\)

1. Show that \(s _ { p } { } ^ { 2 } = 112.6\) to 1 decimal place. Ali claims that there is no difference in the mean arm spans of adult male swimmers and of adult male athletes.
2. Stating your hypotheses clearly, test this claim at the \(10 \%\) level of significance.
   (5) Bea believes that the variances of the arm spans of adult male swimmers and adult male athletes are not equal.
3. Show that, at the \(10 \%\) level of significance, the data support Bea's belief. State your hypotheses and show your working clearly. Ali and Bea combine their work and present their results to their tutor, Clive.
4. Explain why Clive is not happy with their research and state, with a reason, which of the tests in parts (b) and (c) is not valid.