Standard +0.3 This is a standard S4 hypothesis testing question covering Type I/II error definitions, deriving a critical region for a binomial test, and calculating error probabilities. While it requires multiple steps and understanding of error types, all techniques are routine for this specification level with no novel problem-solving required—slightly easier than average due to straightforward application of textbook methods.
a Type II error.
A manufacturer sells socks in boxes of 50 .
The mean number of faulty socks per box is 7.5 . In order to reduce the number of faulty socks a new machine is tried. A box of socks made on the new machine was tested and the number of faulty socks was 2.
(b) (i) Assuming that the number of faulty socks per box follows a binomial distribution derive a critical region needed to test whether or not there is evidence that the new machine has reduced the mean number of faulty socks per box. Use a \(5 \%\) significance level.
Stating your hypotheses clearly, carry out the test in part (i).
(c) Find the probability of the Type I error for this test.
(d) Given that the true mean number of faulty socks per box on the new machine is 5 , calculate the probability of a Type II error for this test.
(e) Explain what would have been the effect of changing the significance level for the test in part (b) to \(2 \frac { 1 } { 2 } \%\).
2. (a) Define
\begin{enumerate}[label=(\roman*)]
\item a Type I error,
\item a Type II error.
A manufacturer sells socks in boxes of 50 .\\
The mean number of faulty socks per box is 7.5 . In order to reduce the number of faulty socks a new machine is tried. A box of socks made on the new machine was tested and the number of faulty socks was 2.\\
(b) (i) Assuming that the number of faulty socks per box follows a binomial distribution derive a critical region needed to test whether or not there is evidence that the new machine has reduced the mean number of faulty socks per box. Use a $5 \%$ significance level.
\item Stating your hypotheses clearly, carry out the test in part (i).\\
(c) Find the probability of the Type I error for this test.\\
(d) Given that the true mean number of faulty socks per box on the new machine is 5 , calculate the probability of a Type II error for this test.\\
(e) Explain what would have been the effect of changing the significance level for the test in part (b) to $2 \frac { 1 } { 2 } \%$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel S4 2006 Q2 [13]}}