7 Jim , a mathematics teacher, knows that the marks, \(X\), achieved by his students can be modelled by a normal distribution with unknown mean \(\mu\) and unknown variance \(\sigma ^ { 2 }\). Jim selects 12 students at random and from their marks he calculates that \(\bar { x } = 64.8\) and \(s ^ { 2 } = 93.0\).

1. 1. An estimate for the standard error of the sample mean is \(d\). Show that \(d ^ { 2 } = 7.75\).
   2. Construct an \(80 \%\) confidence interval for \(\mu\).
2. 1. Write down a confidence interval for \(\mu\), based on Jim's sample of 12 students, which has a width of 10 marks.
   2. Determine the percentage confidence level for the interval found in part (b)(i).