Exercise 1 Investigators are planning to conduct a pilot study to gather information to support the planning of a larger study. The study will involve 50 participants randomized 1:1 to two conditions. They are interested in how precisely they will be able to estimate various quantities.

1. They will use the pilot data to estimate the difference in means between the two conditions using a 95% confidence interval. What is the probability that with this sample size, a 95% confidence interval for the standardized difference between two means would not exceed 0.6?

2. They expect that 80% of the participants will be retained, that is, will complete the study. If they compute a 95% confidence interval for the proportion out of 50 who are retained, what is the expected width of the confidence interval? (Hint: You can use the prec_prop function from the precsize package with the “wilson’’ option.)