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.
The family error rate, also known as the experiment-wise error rate or the family-wise error rate (FWER), refers to the probability of making at least one Type I error (rejecting a true null hypothesis) in multiple hypothesis testing.
In this case, we need to conduct 3 two-sample t-tests at a significance level of α=0.05. Since each test has an individual significance level of α=0.05, the family error rate can be calculated as:
Family Error Rate = 1 - (1 - α)^(Number of Tests)
Family Error Rate = 1 - (1 - 0.05)^3
Family Error Rate ≈ 1 - 0.95^3
Family Error Rate ≈ 1 - 0.857375
Family Error Rate ≈ 0.1426
Therefore, the family error rate for the 3 t-tests is approximately 0.1426 or 14.26%.
To determine the family error rate for the number of t-tests identified in part (a), each conducted at a level of α=0.05, we need to calculate the overall probability of making a Type I error (rejecting a true null hypothesis) in any of the tests.
Since there are 3 pairs of groups (AB, BC, AC) and we are conducting each test at a significance level of α=0.05, the probability of making a Type I error in any single test is 0.05.
To calculate the family error rate, we need to use a method called the Bonferroni correction. The Bonferroni correction adjusts the significance level for each individual test in order to control for the overall family-wise error rate.
The corrected significance level for each individual test is given by α/(number of tests). In this case, α = 0.05 and the number of tests is 3.
So, the corrected significance level for each individual test is 0.05/3 = 0.0167.
Therefore, the family error rate for the number of t-tests identified in part (a) is 0.0167.