7. A random sample of 8 apples is taken from an orchard and the weight, in grams, of each apple is measured. The results are given below.

$$\begin{array} { l l l l l l l l } 
143 & 131 & 165 & 122 & 137 & 155 & 148 & 151
\end{array}$$
\begin{enumerate}[label=(\alph*)]
\item Calculate unbiased estimates for the mean and the variance of the weights of apples.

A population has an unknown mean $\mu$ and an unknown variance $\sigma ^ { 2 }$\\
A random sample represented by $X _ { 1 } , X _ { 2 } , X _ { 3 } , \ldots , X _ { 8 }$ is taken from this population.
\item Explain why $\sum _ { i = 1 } ^ { 8 } \left( X _ { i } - \mu \right) ^ { 2 }$ is not a statistic.

Given that $\mathrm { E } \left( S ^ { 2 } \right) = \sigma ^ { 2 }$, where $S ^ { 2 }$ is an unbiased estimator of $\sigma ^ { 2 }$ and the statistic

$$Y = \frac { 1 } { 8 } \left( \sum _ { i = 1 } ^ { 8 } X _ { i } ^ { 2 } - 8 \bar { X } ^ { 2 } \right)$$
\item find $\mathrm { E } ( Y )$ in terms of $\sigma ^ { 2 }$
\item Hence find the bias, in terms of $\sigma ^ { 2 }$, when $Y$ is used as an estimator of $\sigma ^ { 2 }$\\

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\end{enumerate}

\hfill \mbox{\textit{Edexcel S3 2016 Q7 [9]}}