A random sample \(W _ { 1 } , W _ { 2 } \ldots , W _ { n }\) is taken from a distribution with mean \(\mu\) and variance \(\sigma ^ { 2 }\)
Write down \(\mathrm { E } \left( \sum _ { i = 1 } ^ { n } W _ { i } \right)\) and show that \(\mathrm { E } \left( \sum _ { i = 1 } ^ { n } W _ { i } ^ { 2 } \right) = n \left( \sigma ^ { 2 } + \mu ^ { 2 } \right)\)
An estimator for \(\mu\) is
$$\bar { X } = \frac { 1 } { n } \sum _ { i = 1 } ^ { n } W _ { i }$$
Show that \(\bar { X }\) is a consistent estimator for \(\mu\).
An estimator of \(\sigma ^ { 2 }\) is
$$U = \frac { 1 } { n } \sum _ { i = 1 } ^ { n } W _ { i } ^ { 2 } - \left( \frac { 1 } { n } \sum _ { i = 1 } ^ { n } W _ { i } \right) ^ { 2 }$$
Find the bias of \(U\).
Write down an unbiased estimator of \(\sigma ^ { 2 }\) in the form \(k U\), where \(k\) is in terms of \(n\).
Turn over
George owns a garage and he records the mileage of cars, \(x\) thousands of miles, between services. The results from a random sample of 10 cars are summarised below.
$$\sum x = 113.4 \quad \sum x ^ { 2 } = 1414.08$$
The mileage of cars between services is normally distributed and George believes that the standard deviation is 2.4 thousand miles.
Stating your hypotheses clearly, test, at the \(5 \%\) level of significance, whether or not these data support George’s belief.
2. Every 6 months some engineers are tested to see if their times, in minutes, to assemble a particular component have changed. The times taken to assemble the component are normally distributed. A random sample of 8 engineers was chosen and their times to assemble the component were recorded in January and in July. The data are given in the table below.
\end{table}
Table 1
Figure 1 shows a graph of the power function for the scientist's test.
On the same axes draw the graph of the power function for the statistician's test.
Given that it takes 20 minutes to collect and test a 20 ml sample and 15 minutes to collect and test a 10 ml sample
show that the expected time of the statistician's test is slower than the scientist's test for \(\lambda \mathrm { e } ^ { - \lambda } > \frac { 1 } { 3 }\)
By considering the times when \(\lambda = 1\) and \(\lambda = 2\) together with the power curves in part (e) suggest, giving a reason, which test you would use.
(2)
\begin{figure}[h]
The carbon content, measured in suitable units, of steel is normally distributed. Two independent random samples of steel were taken from a refining plant at different times and their carbon content recorded. The results are given below.
Sample \(A : \quad 1.5 \quad 0.9 \quad 1.3 \quad 1.2\)
\(\begin{array} { l l l l l l l } \text { Sample } B : & 0.4 & 0.6 & 0.8 & 0.3 & 0.5 & 0.4 \end{array}\)
Stating your hypotheses clearly, carry out a suitable test, at the \(10 \%\) level of significance, to show that both samples can be assumed to have come from populations with a common variance \(\sigma ^ { 2 }\).
Showing your working clearly, find the \(99 \%\) confidence interval for \(\sigma ^ { 2 }\) based on both samples.