Problem 1. Suppose you have a coin that is known to be one of two types: either the probability
of heads is p = 1/5 (call that H0) or the probability of heads is p = 3/10 (call that HA). Consider
testing H0 versus HA by Flipping the coin independently a certain number n of times.

a) Write down the usual sample space for n = 5, the probabilities P0 and PA, and the random
variable X ="number of heads". In a table, give the Type I error rate α and the power β for
each decision rule of the form "reject H0 if X > c". (How many values of c do you need to
consider?)

b) If you flip n = 10 times, is there a decision rule with α ≤ 0.1 and β ≥ 0.7? (You'd want that
decision rule to be of the form "reject H0 if #heads > c" why is that again?). If so, what
is it and what are α and β? If not, why not?

c) If you flip n = 100 times, is there a decision rule with α ≤ 0.05 and β ≥ 0.8? Explain.
d) How large should the number n of trials be so that there is decision rule with α ≤ 0.05 while
β ≥ 0.95? Give the decision rule and its α and β.