2022 HCI P2 Q7

A company uses a machine to produce chocolate bars. The machine is designed to produce chocolate bars with average fat content of \(30\)g. After using the machine for many years, the manager wishes to test, at \(5\%\) level of significance, if the machine still maintains the average fat content in the chocolate bars at \(30\)g. He selects a sample of \(40\) chocolate bars for testing.

1. State, giving a reason, whether it is necessary to assume that the fat content of the chocolate bars is normally distributed for the test to be valid.[1]

After some years of usage, the machine broke down. While waiting for the new machine to arrive, the company reverted to their traditional mode of producing handmade chocolate bars. The manager suspects that handmade chocolate bars have higher average fat content than those made by the machines. To verify his suspicion, the manager asked his staff to perform a hypothesis test at \(5\%\) significance level, on a random sample of \(40\) handmade chocolate bars.

The fat content, \(x\) grams, of the \(40\) handmade chocolate bars are summarised as follows:

\({\sum x = 1220,} \qquad \sum(x - 30.5)^2 = 50\)

2. Calculate the unbiased estimates of the population mean and variance of the fat content in handmade chocolate bars.[2]
3. The sample mean fat content of the handmade chocolate bars is denoted by \(\bar{x}\) grams. State the null and alternate hypotheses and calculate the range of values of \(\bar{x}\) for which the null hypothesis would be rejected at \(5\%\) level of significance. Hence conclude if the manager’s suspicion is valid at \(5\%\) significance level.[4]

4. The manager now knows the population variance of the fat content in handmade chocolate bars. It is given that the population variance is smaller than the unbiased estimate of the population variance calculated in part (ii). Without carrying out another hypothesis test, explain with justification, if there will be a change in the manager’s conclusion about his suspicion at \(5\%\) significance level.

   [1]