2013 AJC P2 Q9

Mr Gru claims that the average height of his minions is \(k\) cm. A random sample of \(100\) minions is chosen and the height, \(X\) cm of each minion is measured. It is found that \(\sum{\left( x-50 \right)=-96,}\) and \(\sum{{{\left( x-50 \right)}^{2}}=1879.}\)

1. Find unbiased estimates of the mean and variance of \(X\).[2]
2. What do you understand by the term “unbiased estimate”?[1]

Based on this sample, a hypothesis test is carried out and it was found that there is significant evidence at \(5\%\) significance level that Mr Gru had underestimated the average height of his minions.

3. Find an inequality satisfied by \(k\).[3]
4. Is there a need to make the assumption that the height of a randomly chosen minion is normally distributed? Justify your answer.[1]