2018 CJC P2 Q10

In a large busy restaurant, the mean time taken for a server to clear a table and set it for the next guest is \(4.5\) minutes. In order to improve the quality of service and maintain a clean environment, the restaurant manager introduced a new routine to clear tables. A random sample of \(30\) servers is taken and the time taken by each server to clear a table, \(x\) minutes, is recorded. The data are summarised as follows:

\(\sum{x=132}\) and \(\sum{{{x}^{2}}=\mathrm{583}\mathrm{.96}}\).

1. Find unbiased estimates of the population mean and variance.[2]
2. Test, at the \(10\%\) significance level, whether the mean time taken for a server to clear a table has changed.[5]
3. Using the results in part (ii), state the range of values of \(\alpha \), where \(\alpha \%\) is the level of significance, at which a test would lead the manager to conclude that the mean time has not changed.[1]
4. Without carrying out another test, state with reasons, the conclusion if the test at \(5\%\) significance level was to determine whether the mean time taken to clear a table is less than \(4.5\) minutes.[2]

After a few weeks, based on feedback from servers and guests, the manager modifies the routine. Another random sample of \(30\) servers is taken and their time taken to clear a table has mean \(\bar{x}\) minutes and variance \(1.01\) minutes\(^{2}\).

5. Find the set of values of \(\bar{x}\) for which a test at \(1\%\) significance level concludes that the mean time taken to clear a table is greater than \(4.5\) minutes, giving your answer to \(3\) decimal places.[4]