6. A company claims that a quarter of the bolts sent to them are faulty. To test this claim the number of faulty bolts in a random sample of 50 is recorded.
\begin{enumerate}[label=(\alph*)]
\item Give two reasons why a binomial distribution may be a suitable model for the number of faulty bolts in the sample.
\item Using a 5\% significance level, find the critical region for a two-tailed test of the hypothesis that the probability of a bolt being faulty is $\frac { 1 } { 4 }$. The probability of rejection in either tail should be as close as possible to 0.025
\item Find the actual significance level of this test.

In the sample of 50 the actual number of faulty bolts was 8 .
\item Comment on the company's claim in the light of this value. Justify your answer.

The machine making the bolts was reset and another sample of 50 bolts was taken. Only 5 were found to be faulty.
\item Test at the $1 \%$ level of significance whether or not the probability of a faulty bolt has decreased. State your hypotheses clearly.
\end{enumerate}

\hfill \mbox{\textit{Edexcel S2  Q6 [15]}}