5. A large company has designed an aptitude test for new recruits. The score, \(S\), for an individual taking the test, has a normal distribution with mean \(\mu\) and standard deviation \(\sigma\). In order to estimate \(\mu\) and \(\sigma\), a random sample of 15 new recruits were given the test and their scores, \(x\), are summarised as $$\sum x = 880 \quad \sum x ^ { 2 } = 54892$$

1. Calculate a 95\% confidence interval for
   1. \(\mu\),
   2. \(\sigma\). The company wants to ensure that no more than \(80 \%\) of new recruits pass the test.
2. Using values from your confidence intervals in part (a), estimate the lowest pass mark they should set.