When np0 < 10 or n(1 − p0) < 10, we cannot use the normal approximation, but we can use the binomial distribution to perform what is known as an exact test. Let p be the probability that a given coin lands heads. The coin is tossed 10 times and comes up heads 9 times. Test H0: p = 0.5 versus H1: p > 0.5, as follows.

a. Let n be the number of tosses and let X denote the number of heads. Find the values of n and X in this example.

b. The distribution of X is binomial. Assuming H0 is true, find n and p.

c. Because the alternate hypothesis is p > 0.5, large values of X support H1. Find the probability of observing a value of X as extreme as or more extreme than the value actually observed, assuming H0 to be true. This is the P-value.

d. Do you reject H0 at the α = 0.05 level?