2013 IJC P2 Q11 [Modified]

In a shopping mall, the number of hours that customers spend shopping in the mall follows a normal distribution. The manager of the mall claims that the average number of hours customers spend in the mall is at least \(2\) hours. To test this claim, a random sample of \(60\) customers is taken and the time, \(x\) hours, spent is measured. The results are summarized by \(\sum{x}=110.5\) and \(\sum{{{x}^{2}}}=220.3\).

1. Find unbiased estimates of the population mean and variance.[3]
2. Test at the \(2\%\) significance level whether the manager's claim is valid.[4]
3. Explain, in the context of the question, the meaning of “at the \(2\%\) significance level”.[1]

After an upgrading project to improve the facilities in the mall, a new random sample of \(55\) customers is taken. The data gives a sample mean of \(\bar{x}\) with sample standard deviation \(0.51\) hours. A test at the \(2\%\) significance level indicates that the manager’s claim is valid with the improved facilities.

4. Using an algebraic method, calculate the range of values of \(\bar{x}\).[5]