A bank categorizes its customers into one of three groups based on their banking habits. A random sample of 30 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.
Group n x¯ s
A 30 48 7
B 30 51 8
C 30 54 10
The bank manager will investigate whether there is a significant difference in mean numbers of bank visits for the groups. Multiple two-sample t -tests will be conducted, each at the significance level of α=0.05 .

(a) How many t -tests will need to be conducted for the manager’s investigation? List the pairs of groups for each test.

The manager wants to know the probability of making a Type I error for multiple t -tests, not just for a single t -test.

(b) 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.

(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.

(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.