A common tool in laboratory medicine and biology is the microarray, a device that can measure the expression levels of thousands of genes at once. So, for example, an investigator might collect samples from normal subjects and subjects with cancer in the hopes of finding genes that are significantly associated with cancer. The investigator is therefore testing a null hypothesis for every gene on the array. Suppose that a given array has 2,000 genes, of which 20 are known to be associated with cancer. Suppose further that the investigator's hypothesis tests have a Type I error rate of 5% and a Type II error rate of 20%.

(a) In words, what are the null and alternative hypotheses for each test on a single gene.

(b) Out of the 2,000 hypothesis tests that the investigator carries out, how many are type I errors?

(c) How many are type II errors? (d) How many times did the investigator correctly reject the null hypothesis?

(e) What was the investigator's false discovery rate? (i.e., the fraction of null hypothesis rejections that were incorrect)

(f) If, for each gene, a 95% confidence interval was calculated for the association between the gene and cancer status, how many of those confidence intervals would contain the true association for that gene?