N2009 P2 Q10

A company supplies sugar in small packets. The mass of sugar in one packet is denoted by \(X\) grams. The masses of a random sample of 9 packets are summarised by \(\sum x=86.4\), \(\sum x^2=835.92\).

1. Calculate unbiased estimates of the mean and variance of \(X\).[2]

The mean mass of sugar in a packet is claimed to be 10 grams. The company directors want to know whether the sample indicates that this claim is incorrect.

2. Stating a necessary assumption, carry out a t-test at the 5% significance level. Explain why the Central Limit Theorem does not apply in this context.[7]
3. Suppose now that the population variance of \(X\) is known, and that the assumption made in part (ii) is still valid. What change would there be in carrying out the test?[1]