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.
$$\begin{array} { l l l l l l l l }
143 & 131 & 165 & 122 & 137 & 155 & 148 & 151
\end{array}$$
- 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. - 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
$$Y = \frac { 1 } { 8 } \left( \sum _ { i = 1 } ^ { 8 } X _ { i } ^ { 2 } - 8 \bar { X } ^ { 2 } \right)$$
- find \(\mathrm { E } ( Y )\) in terms of \(\sigma ^ { 2 }\)
- Hence find the bias, in terms of \(\sigma ^ { 2 }\), when \(Y\) is used as an estimator of \(\sigma ^ { 2 }\)