Don’t do what Donny Doesn’t do. Donny Don’t wants to see if eating jelly beans of a certain color is associated with acne.

1. (a) Donny Don’t pick 20 different colors and tests at level 0.05 the null hypothesis of no association for each of the 20 colors. The 20 tests are based on 20 independent samples, one sample used to perform each test. The probability that at least one true null hypothesis is falsely rejected is called the “family-wise error rate (FWER)”. Compute the family-wise error rate of Donny’s tests.

2. (b) Now Donny picks a color, selects a sample, and uses the data to test at level 0.05 the null hypothesis of no association. He continues in this manner until he obtains a sample for which the null hypothesis of no association is rejected, and then he stops testing.

   1. Find the FWER.

   2. Suppose that there really is no association between acne and eating jelly beans of any color. Let N be the number of tests that Donny performs, including the test where he rejects the null hypothesis. Compute EH0(N) and
      PH0(N ≤ E(N)).

3. (c) Explain to Donny Don’t why he shouldn’t do what he did. Try to make your explanation as non-technical as possible, so that someone without a background in statistics can understand.