2018 MJC P2 Q9

Based on past records, shoppers spend an average of \(1.5\) hours at MJ mall per visit. After a revamp and expansion to include more stores, the mall management wishes to conduct a hypothesis test to check if the duration that shoppers spend in the mall per visit has increased. A random sample of \(50\) shoppers were surveyed and the duration of time, \(x\), in hours per visit is recorded. The sample sum is \(82.5\) hours and the sample variance is \(0.425\) hours\(^{2}\).

1. State appropriate hypotheses for the test and find the unbiased estimates of the population mean and variance.[3]
2. Explain why the mall management is able to carry out a hypothesis test without making any assumptions about the distribution of the duration shoppers spend at MJ mall per visit.[1]
3. Calculate the \(p\)-value and state its meaning in context of the question. Hence state the set of values of the level of significance \(\alpha \%\) for which the management’s hypothesis is valid.[4]
4. The mall management believes that the mean spending per shopper per visit is \(\$150\). A test at \(5\%\) level of significance found that there is significant evidence that the population mean spending is less than \(\$150\). Using only this information, explain if the following statements are necessarily true, necessarily false, or neither necessarily true nor necessarily false.
   1. There is significant evidence at the \(10\%\) level of significance that the population mean spending is less than \(\$150\).
   2. There is significant evidence at the \(5\%\) level of significance that the population mean spending differs from \(\$150\).[2]