An insurance company believes that claim amounts in a certain portfolio of policies follow a normal distribution. An analyst chose 61 policies at random which gave a sample mean of K5,230 and a sample standard deviation of K810.

(a) Construct a 95% confidence interval for the mean claim amount in the portfolio.

(b) The company has changed its loss assessment processes in order to reduce claim sizes on average, targeting a reduction of K200 compared to the current mean. It does not expect a change to the variability of claim amounts. The company intends to verify whether the target has been met by using a sample of claims to test the null hypothesis that there is no change, against a one-sided alternative hypothesis. Company policy is to perform statistical tests at a significance level of 5%.

Determine the smallest number of claims that would need to be sampled under the new processes for a K200 reduction to be statistically significant in the test.

(a) In a survey conducted by a mail order company a random sample of 200 customers yielded 172 who indicated that they were highly satisfied with the delivery time of their orders. Calculate an approximate 95% confidence interval for the proportion of the company's customers who are highly satisfied with delivery times.

(b) A random sample of size 49 from a normal distribution gives a 99% confidence interval for the population mean as (30, 50). Find a 90% confidence interval for the population mean based on this information.