2 Pat makes and sells fruit cakes at a local market. On her stall a sign states that the average weight of the cakes is 1 kg . A trading standards officer carries out a routine check of a random sample of 8 of Pat's cakes to ensure that they are not underweight, on average. The weights, in kg , that he records are as follows.
$$\begin{array} { l l l l l l l l }
0.957 & 1.055 & 0.983 & 0.917 & 1.015 & 0.865 & 1.013 & 0.854
\end{array}$$
- On behalf of the trading standards officer, carry out a suitable test at a \(5 \%\) level of significance, stating your hypotheses clearly. Assume that the weights of Pat's fruit cakes are Normally distributed.
- Find a 95\% confidence interval for the true mean weight of Pat's fruit cakes.
Pat's husband, Tony, is the owner of a factory which makes and supplies fruit cakes to a large supermarket chain. A large random sample of \(n\) of these cakes has mean weight \(\bar { x } \mathrm {~kg}\) and variance \(0.006 \mathrm {~kg} ^ { 2 }\).
- Write down, in terms of \(n\) and \(\bar { x }\), a \(95 \%\) confidence interval for the true mean weight of cakes produced in Tony's factory.
- What is the size of the smallest sample that should be taken if the width of the confidence interval in part (iii) is to be 0.025 kg at most?