7 A random sample of 50 full-time university employees was selected as part of a higher education salary survey. The annual salary in thousands of pounds, \(x\), of each employee was recorded, with the following summarised results. $$\sum x = 2290.0 \quad \text { and } \quad \sum ( x - \bar { x } ) ^ { 2 } = 28225.50$$ Also recorded was the fact that 6 of the 50 salaries exceeded \(\pounds 60000\).

1. 1. Calculate values for \(\bar { x }\) and \(s\), where \(s ^ { 2 }\) denotes the unbiased estimate of \(\sigma ^ { 2 }\).
   2. Hence show why the annual salary, \(X\), of a full-time university employee is unlikely to be normally distributed. Give numerical support for your answer.
2. 1. Indicate why the mean annual salary, \(\bar { X }\), of a random sample of 50 full-time university employees may be assumed to be normally distributed.
   2. Hence construct a \(99 \%\) confidence interval for the mean annual salary of full-time university employees.
3. It is claimed that the annual salaries of full-time university employees have an average which exceeds \(\pounds 55000\) and that more than \(25 \%\) of such salaries exceed £60000. Comment on each of these two claims.