Explain the meaning of the following terms in the context of hypothesis testing: Type I error, Type II error, operating characteristic, power.
A market research organisation is designing a sample survey to investigate whether expenditure on everyday food items has increased in 2011 compared with 2010. For one of the populations being studied, the random variable \(X\) is used to model weekly expenditure, in \(\pounds\), on these items in 2011, where \(X\) is Normally distributed with mean \(\mu\) and variance \(\sigma ^ { 2 }\). As the corresponding mean value in 2010 was 94 , the hypotheses to be examined are
$$\begin{aligned}
& \mathrm { H } _ { 0 } : \mu = 94
& \mathrm { H } _ { 1 } : \mu > 94
\end{aligned}$$
By comparison with the corresponding 2010 value, \(\sigma ^ { 2 }\) is assumed to be 25 .
The following criteria for the survey are laid down.
If in fact \(\mu = 94\), the probability of concluding that \(\mu > 94\) must be only \(2 \%\)
If in fact \(\mu = 97\), the probability of concluding that \(\mu > 94\) must be \(95 \%\)
A random sample of size \(n\) is to be taken and the usual Normal test based on \(\bar { X }\) is to be used, with a critical value of \(c\) such that \(\mathrm { H } _ { 0 }\) is rejected if the value of \(\bar { X }\) exceeds \(c\). Find \(c\) and the smallest value of \(n\) that is required.
Sketch the power function of an ideal test for examining the hypotheses in part (ii).