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.
- Give two reasons why a binomial distribution may be a suitable model for the number of faulty bolts in the sample.
- 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
- Find the actual significance level of this test.
In the sample of 50 the actual number of faulty bolts was 8 .
- 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.
- Test at the \(1 \%\) level of significance whether or not the probability of a faulty bolt has decreased. State your hypotheses clearly.