2015 AJC P2 Q10

A farmer claims that the mean weight of the melons grown in his farm is at least \(1.2\) kg. A random sample of \(10\) melons is chosen and the weight, \(x\) kg, of each melon is recorded. The results are as follows:

\[ \begin{matrix} 1.037 \quad & 0.914 \quad & 1.019 \quad & 1.234 \quad & 1.110 \\ 1.417 \quad & 1.008 \quad & 0.846 \quad & 1.105 \quad & 1.331 \end{matrix} \]

1. By combining the \(2\) sets of data, find unbiased estimates of the population mean and population variance of the weight of a melon grown by this farmer.[2]
2. The farmer decides to change the claim to “the mean weight of the melons grown in his farm is at least \(m\) kg”. Based on the combined sample of size \(50\), find the set of values of \(m\) for which the farmer’s claim is not rejected at the \(5\%\) significance level.[3]