Exercise 1 Investigators have a new technique for performing gallbladder surgery. They need your help designing a single-arm study to assess the new technique. If its success rate is \(45\%\) or less, the new technique would not be acceptable for use. They plan a single-arm study to test \(H_0 \colon p \leq 0.45\) versus \(H_A \colon p > 0.45\). They want a type I error rate of not higher than 0.05 and type II error rate not higher than 0.2.

1. If they expect that the true success rate will be \(95\%\), would a sample of 4 patients be sufficiently large to achieve their objectives using a single-stage design? Fill in the following table using the binomial probability mass function and explain how the table enables you to answer this question.

2. For the previous question, find the optimal and minimax Simon two-stage designs (use software). Explain how to calculate the probability of early termination under the null (PET(\(p_0\))) and the expected sample size under the null (EN(\(p_0\))).

3. Suppose that they expect the true success rate to be \(65\%\). If they use a single-stage design, what is the minimum sample size required and the critical value? If they use a Simon two-stage design, what are the optimal and minimax designs? What are the PET and the expected sample sizes when the null is true? Briefly discuss how the three designs compare to each other.