An experimenter has prepared a drug dosage level that she claims will induce sleep for 80% of people suffering from insomnia. After examining the dosage, we feel that her claims regarding the effectiveness of the dosage are inflated. In an attempt to disprove her claim, we administer her prescribed dosage to 20 insomniacs and we observe Y, the number for whom the drug dose induces sleep. We wish to test the hypothesis H₀: p 0.8 vs. the alternative H_A: p < 0.8. Assume that the rejection region { y 12 } is used.

1. In terms of this problem, what is a type I error?
2. Find .
3. In terms of this problem, what is a type II error?
4. Find when p = 0.6.
5. Find when p = 0.4.

A two-stage clinical trial is planned for testing H₀: p 0.10 vs. H_A: p > 0.10, where p is the proportion of responders among patients who were treated by the protocol treatment. At the first stage, 15 patients are accrued and treated. If 4 or more responders are observed among the (first) 15 patients, H₀ is rejected, the study is terminated, and no more patients are accrued. Otherwise, another 15 patients will be accrued and treated in the second stage. If a total of 6 or more responders are observed among the 30 patients accrued in the two stages (15 in the first stage and 15 more in the second stage), then H₀ is rejected. For example, if 5 responders are found among the first-stage patients, H₀ is rejected and the study is over. However, if 2 responders are found among the first-stage patients, 15 second-stage patients are accrued, and an additional 4 or more responders (for a total of 6 or more among the 30) are identified, H₀ is rejected and the study is over.

1. Use the binomial table to find the numerical value of for this testing procedure.
2. Use the binomial table to find the probability of rejecting the null hypothesis when using this rejection region if p = 0.30.
3. For the rejection region defined above, find if p = 0.30.