5. Past records show that the proportion of customers buying organic vegetables from Tesson supermarket is 0.35 During a particular day, a random sample of 40 customers from Tesson supermarket was taken and 18 of them bought organic vegetables.

1. Test, at the \(5 \%\) level of significance, whether or not this provides evidence that the proportion of customers who bought organic vegetables has increased. State your hypotheses clearly. The manager of Tesson supermarket claims that the proportion of customers buying organic eggs is different from the proportion of those buying organic vegetables. To test this claim the manager decides to take a random sample of 50 customers.
2. Using a \(5 \%\) level of significance, find the critical region to enable the Tesson supermarket manager to test her claim. The probability for each tail of the region should be as close as possible to \(2.5 \%\) During a particular day, a random sample of 50 customers from Tesson supermarket is taken and 8 of them bought organic eggs.
3. Using your answer to part (b), state whether or not this sample supports the manager's claim. Use a \(5 \%\) level of significance.
4. State the actual significance level of this test. The proportion of customers who buy organic fruit from Tesson supermarket is 0.2 During a particular day, a random sample of 200 customers from Tesson supermarket is taken. Using a suitable approximation, the probability that fewer than \(n\) of these customers bought organic fruit is 0.0465 correct to 4 decimal places.
5. Find the value of \(n\).