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

(b) A t-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 Type I error; if the null hypothesis is not rejected, the result is not statistically significant, which would not be a Type I error. Let S represent a statistically significant result, and let N represent a result that is not statistically significant.

(i) If P(S)=0.05, what is the value of P(N) ?

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

The figure presents a tree diagram. Starting at the left side, there are two branches labeled N and S. Branches N and S each split into two additional branches, labeled N and S.
(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 two t-tests, each conducted at a level of α=0.05. Show your work.

Question 3
(c) Determine the family error rate for the number of t-tests identified in part (a), each conducted at a level of α=0.05. Show your work.

Question 4
(d) The manager wants the family error rate to be close to 0.05 . Suggest a single significance level α that could be used for all of the individual t -tests that will bring the family error rate close to 0.05 . Show work to support your suggested level.