2024 DHS H1 Q10

A chocolate company produces chocolate bars. It is known that the mean mass of the chocolate bars is \(30\) grams. The manager decides to take a random sample of \(50\) chocolate bars from those produced for quality control purposes.

The masses, \(x\) grams, of the random sample of \(50\) chocolate bars are summarised as follows.

\[\sum(x-30)=35.5 \quad \sum(x-30)^2=360.43\]

1. State what it means for a sample to be random in this context.[1]
2. Test, at the \(2\)% level of significance, the claim that the mean mass of the chocolate bars is \(30\) grams. You should state your hypotheses and define any symbols that you use.[6]
3. State, to the nearest percent, the smallest level of significance at which the null hypothesis will be rejected using the given set of data.[1]

The manager modifies the machine to improve quality. After the machine is modified, the mean mass of the chocolate bars is \(\mu\) grams and the variance of the masses of the chocolate bars is \(0.7\) grams\({}^2\). Another random sample of \(40\) chocolate bars are taken. A significance test of the null hypothesis \(\mu=30\) against the alternative hypothesis \(\mu>30\) is carried out at the \(5\)% significance level.

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