6 A supermarket buys pears from a local supplier. The supermarket requires the mean weight of the pears to be at least 175 grams. William, the fresh-produce manager at the supermarket, suspects that the latest batch of pears delivered does not meet this requirement.
\begin{enumerate}[label=(\alph*)]
\item William weighs a random sample of 6 pears, obtaining the following weights, in grams.

$$\begin{array} { l l l l l l } 
160.6 & 155.4 & 181.3 & 176.2 & 162.3 & 172.8
\end{array}$$

Previous batches of pears have had weights that could be modelled by a normal distribution with standard deviation 9.4 grams. Assuming that this still applies, show that a hypothesis test at the $5 \%$ level of significance supports William's suspicion.\\
(7 marks)
\item William then weighs a random sample of 20 pears. The mean of this sample is 169.4 grams and $s = 11.2$ grams, where $s ^ { 2 }$ is an unbiased estimate of the population variance.

Assuming that the population from which this sample is taken has a normal distribution but with unknown standard deviation, test William's suspicion at the $\mathbf { 1 \% }$ level of significance.
\item Give a reason why the probability of a Type I error occurring was smaller when conducting the test in part (b) than when conducting the test in part (a).
\end{enumerate}

\hfill \mbox{\textit{AQA S2 2013 Q6 [13]}}