5 In 2001, the mean height of students at the end of their final year at Bright Hope Secondary School was 165 centimetres. In 2010, David and James selected a random sample of 100 students who were at the end of their final year at this school. They recorded these students' heights, \(x\) centimetres, and found that \(\bar { x } = 167.1\) and \(s ^ { 2 } = 101.2\). To investigate the claim that the mean height had increased since 2001, David and James each correctly conducted a hypothesis test. They used the same null hypothesis and the same alternative hypothesis. However, David used a \(5 \%\) level of significance whilst James used a \(1 \%\) level of significance.

1. 1. Write down the null and alternative hypotheses that both David and James used.
      (l mark)
   2. Determine the outcome of each of the two hypothesis tests, giving each conclusion in context.
   3. State why both David and James made use of the Central Limit Theorem in their hypothesis tests.
2. It was later found that, in 2010, the mean height of students at the end of their final year at Bright Hope Secondary School was actually 165 centimetres. Giving a reason for your answer in each case, determine whether a Type I error or a Type II error or neither was made in the hypothesis test conducted by:
   1. David;
   2. James.