Each of a group of 20 intermediate tennis players is given two rackets, one having nylon strings and the other synthetic gut strings. After several weeks of playing with the two rackets, each player will be asked to state a preference for one of the two types of strings. Let p denote the proportion of all such players who would prefer gut to nylon, and let X be the number of players in the sample who prefer gut. Because gut strings are more expensive, consider the null hypothesis that at most 50% of all such players prefer gut. We simplify this to H0: p = .5, planning to reject H0 only if sample evidence strongly favors gut strings.
a. Is a significance level of exactly .05 achievable? If not, what is the largest α smaller than .05 that is achievable?
b. If 60% of all enthusiasts prefer gut, calculate the probability of a type II error using the significance level from part (a). Repeat if 80% of all enthusiasts prefer gut.
c. If 13 out of the 20 players prefer gut, should H0 be rejected using the significance level of (a)?