2024 SAJC H1 Q12

The branch manager of a bank claims that the mean waiting time for customers to be served by a bank consultant is 12.4 minutes. To investigate this claim, the branch manager plans to conduct a survey on a sample of customers an hour before the bank's closing time.

1. Explain why this method of sampling will not give a random sample for the purpose of the investigation.[1]

After the survey was conducted, the branch manager realised that the waiting time for each customer before they were served at the counter was of a major concern. He then suspected that the waiting time per customer could be longer than 12.4 minutes. He decided to record the waiting time, \(t\) minutes, for 50 customers at the branch.

The results are summarised by

\[\sum (t - 12.4) = 5.5\ ,\ \sum (t - 12.4)^2 = 10.5\]

2. Find the unbiased estimates of the population mean and variance.[2]
3. Test, at the 5% significance level, whether this set of data supports the branch manager's suspicion[4]
4. Explain what is meant by the \(p\)-value obtained in part (c) in the context of the question.[1]

The branch manager reviewed some processes for the counter service at his branch. A different random sample of 50 customers was chosen and it was found that the mean waiting time for each customer before they were served at the counter was \(k\) minutes.

A significance test of the null hypothesis \(\mu = 12.4\) against the alternative hypothesis \(\mu \neq 12.4\) is carried out at 4% level of significance. It is assumed that the unbiased estimate of the population variance remains the same as in part (b),

5. Find the set of possible values of the sample mean that would lead to the null hypothesis being rejected.[4]