2023 EJC P2 Q11

The amount customers spend is often correlated with the amount of time they spend in stores. According to a national survey of $500,000\( shopping trips, the average time customers spend in a clothing store is \)30$ min. The first, second and third quartiles of the time spent, in min, are also given as follows

1. Explain why the data suggests that shopping times do not follow a normal distribution.[1]

Jack, a clothing store manager, suspects that customers spend less time in his store than the national average.

2. Determine whether Jack should carry out a \(1\)-tail or a \(2\)-tail test. State the hypotheses for the test and define any symbols you use in this context.[3]

Jack collected a random sample of \(50\) shopping times. The results, \(x\) (in min), are summarised as follows:

\(\sum{x=1475}\), \(\sum{{{x}^{2}}=45029}\)

3. Test, at the \(10\%\) significance level, whether Jack’s suspicion is valid.[5]
4. Explain what is meant by ‘at the \(10\%\) significance level’ in the context of the question.[1]
5. Explain why in performing the hypothesis test, there is no need for Jack to be concerned that the shopping times of customers in his store do not follow a normal distribution.[1]

Upon closer inspection of the data, Jack realises that he has made some miscalculations. While there is no mistake in the calculation of \(\sum{x}\), the value of \(\sum{{{x}^{2}}}\) should be larger than initially calculated.

6. Does this miscalculation affect Jack’s initial conclusion? Explain your answer.[2]