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.
Groupn 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.
3 - 2 sample t tests - AB, BC, AC??
(b) 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.
i have no idea :D
I need answers
In this case, we need to conduct multiple two-sample t-tests for each pair of groups. Since there are three groups (A, B, and C), we will need to conduct three pairwise comparisons: AB, BC, and AC.
To determine the family error rate, we need to account for the possibility of making a type I error (rejecting a true null hypothesis) in any of the conducted t-tests.
The family error rate can be calculated using the formula:
Family error rate = 1 - (1 - α)^m
Where:
- α is the significance level for each individual t-test (in this case, 0.05).
- m is the total number of tests conducted (in this case, 3 pairwise comparisons).
Plugging in the values, we get:
Family error rate = 1 - (1 - 0.05)^3
= 1 - (0.95)^3
= 1 - 0.857375
= 0.142625
Therefore, the family error rate for the number of t-tests identified in part (a), each conducted at a level of α = 0.05, is approximately 0.142625 or 14.26%.
(a) The bank manager will need to conduct a total of three t-tests for this investigation. Each t-test will compare the means of two groups. The pairs of groups for each test are as follows:
1. Group A vs Group B: This test compares the mean number of bank visits for Group A and Group B.
2. Group B vs Group C: This test compares the mean number of bank visits for Group B and Group C.
3. Group A vs Group C: This test compares the mean number of bank visits for Group A and Group C.
Therefore, the pairs of groups for each t-test are AB, BC, and AC.
(b) The family error rate for the number of t-tests conducted at a level of α=0.05 can be calculated using the Bonferroni correction. The family error rate is the overall probability of making at least one Type I error (rejecting a null hypothesis when it is true) across all the t-tests.
To calculate the family error rate, we divide the desired significance level (α = 0.05) by the number of tests (3 in this case). Therefore, the family error rate for each t-test will be:
Family error rate = α / Number of tests
= 0.05 / 3
= 0.0167 (rounded to four decimal places)
So, each t-test conducted at a level of α=0.05 will have a family error rate of 0.0167.