6. Past information at a computer shop shows that \(40 \%\) of customers buy insurance when they purchase a product. In a random sample of 30 customers, \(X\) buy insurance.
- Write down a suitable model for the distribution of \(X\).
- State an assumption that has been made for the model in part (a) to be suitable.
The probability that fewer than \(r\) customers buy insurance is less than 0.05
- Find the largest possible value of \(r\).
A second random sample, of 100 customers, is taken.
The probability that at least \(t\) of these customers buy insurance is 0.938 , correct to 3 decimal places. - Using a suitable approximation, find the value of \(t\).
The shop now offers an extended warranty on all products. Following this, a random sample of 25 customers is taken and 6 of them buy insurance.
- Test, at the \(10 \%\) level of significance, whether or not there is evidence that the proportion of customers who buy insurance has decreased. State your hypotheses clearly.