2024 EJC H1 Q10

The owner of an orchard that produces peaches claims that the mean mass of the peaches in his orchard is \(250\) g. The masses of peaches are known to have a normal distribution with variance \(\sigma^2\). A random sample of \(50\) peaches is selected and found to have a sample mean of \(252\) g.

Using this information, a hypothesis test of the null hypothesis \(\mu = 250\) against the alternative hypothesis \(\mu > 250\) is carried out at the \(5\)% significance level. The test shows that there is insufficient evidence to reject the null hypothesis.

1. State in context what \(\mu\) represents and find the range of possible values of \(\sigma^2\).[5]

The orchard also produces nectarines. The owner claims that the mean mass of nectarines is \(160\) g. The distribution of the masses of nectarines has a standard deviation of \(5.8\) g. A customer decides to test the owner's claim and bought a random sample of \(8\) nectarines. The masses of the nectarines, in g, are as follows.

\[158 \quad 152 \quad 161 \quad 154 \quad 155 \quad 151 \quad 161 \quad 160\]

2. Test, at the \(10\)% significance level, whether the owner's claim is valid. State an assumption to be made about the distribution of the masses of nectarines.[5]