Show all your work. Indicate clearly the methods you use, because you will be scored on the correctness of your methods as well as on the accuracy and completeness of your results and explanations.

A bank categorizes its customers into one of three groups based on their banking habits. A random sample of30

customers from each group was selected, and the number of times each customer visited the bank during the past year was recorded. The following table shows the summary statistics.

The bank manager will investigate whether there is a significant difference in mean numbers of bank visits for the groups. Multiple two-samplet

-tests will be conducted, each at the significance level of=0.05

The significance level() of a single hypothesis test is the probability of making a TypeI error. The manager wants to know the probability of making a TypeI error for multiplet-tests, not just for a singlet-test. This probability is called the family error rate for TypeI error, which is also known as the family error rate.

(b) At-test has two possible outcomes: reject or do not reject the null hypothesis. Suppose the null hypothesis is true. If the null hypothesis is rejected, the result is statistically significant, which would be a TypeI error; if the null hypothesis is not rejected, the result is not statistically significant, which would not be a TypeI error. LetS represent a statistically significant result, and letN represent a result that is not statistically significant.

(i) IfP(S)=0.05, what is the value ofP(N)?

The bank manager knows that the investigation will involve conducting multiple two-samplet-tests. The manager begins the investigation by considering two differentt-tests as independent, successive trials. The possible outcomes of the trials,N orS, are shown in the following tree diagram.

(ii) The family error rate is the probability of obtaining a significant result for at least one of the t-tests conducted, under the assumption that the null hypothesis is true. Use the tree diagram to determine the family error rate for twot-tests, each conducted at a level of=0.05. Show your work.

(c) Determine the family error rate for the number of t-tests identified in part (a), each conducted at a level of=0.05. Show your work.

(d) The manager wants the family error rate to be close to 0.05. Suggest a single significance level that could be used for all of the individualt-tests that will bring the family error rate close to0.05. Show work to support your suggested level.