3. Star Farm produces duck eggs. Xander takes a random sample of 20 duck eggs from Star Farm and their widths, $x \mathrm {~cm}$, are recorded. Xander's results are summarised as follows.

$$\sum x = 92.0 \quad \sum x ^ { 2 } = 433.4974$$
\begin{enumerate}[label=(\alph*)]
\item Calculate unbiased estimates of the mean and the variance of the width of duck eggs produced by Star Farm.

Yinka takes an independent random sample of 30 duck eggs from Star Farm and their widths, $y \mathrm {~cm}$, are recorded. Yinka's results are summarised as follows.

$$\sum y = 142.5 \quad \sum y ^ { 2 } = 689.5078$$
\item Treating the combined sample of 50 duck eggs as a single sample, estimate the standard error of the mean.\\
(5)

Research shows that the population of duck egg widths is normally distributed with standard deviation 0.71 cm .

The farmer claims that the mean width of duck eggs produced by Star Farm is greater than 4.5 cm .
\item Using your combined mean, test, at the $5 \%$ level of significance, the farmer's claim. State your hypotheses clearly.
\end{enumerate}

\hfill \mbox{\textit{Edexcel S3 2018 Q3 [13]}}