Each of a group of 20 intermediate tennis players is given two tennis rackets, one 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 $H _ { 0 } : p = .5$ planning to reject $H _ { 0 }$ only if sample evidence strongly favors gut strings.
a. Which of the rejecting regions { 15, 16, 17, 18, 19, 20}, {0, 1, 2, 3, 4, 5}, or { 0, 1, 2, 3, 17, 18, 19, 20} is most appropriate, and why are the other two not appropriate?
b. What is the probability of a type I error for the chosen region of part (a)? Does the region specify a level .05 test? Is it the best level .05 test?
c. If 60% of all enthusiasts prefer gut, calculate the probability of a type II error using the appropriate region from part (a). Repeat if 80% of all enthusiasts prefer gut.
d. If 13 out of the 20 players prefer gut, should $H _ { 0 }$
be rejected using a significance level of .10?

Question

Each of a group of 20 intermediate tennis players is given two tennis rackets, one 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 $H _ { 0 } : p = .5$ planning to reject $H _ { 0 }$ only if sample evidence strongly favors gut strings. 
a. Which of the rejecting regions { 15, 16, 17, 18, 19, 20}, {0, 1, 2, 3, 4, 5}, or { 0, 1, 2, 3, 17, 18, 19, 20} is most appropriate, and why are the other two not appropriate?
b. What is the probability of a type I error for the chosen region of part (a)? Does the region specify a level .05 test? Is it the best level .05 test?
c. If 60% of all enthusiasts prefer gut, calculate the probability of a type II error using the appropriate region from part (a). Repeat if 80% of all enthusiasts prefer gut.
d. If 13 out of the 20 players prefer gut, should $H _ { 0 }$ 
be rejected using a significance level of .10?

Quizplus · Accepted Answer

The Answer of Each of a group of 20 intermediate tennis players is given two tennis rackets, one 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 $H _ { 0 } : p = .5$ planning to reject $H _ { 0 }$ only if sample evidence strongly favors gut strings. 
a. Which of the rejecting regions { 15, 16, 17, 18, 19, 20}, {0, 1, 2, 3, 4, 5}, or { 0, 1, 2, 3, 17, 18, 19, 20} is most appropriate, and why are the other two not appropriate?
b. What is the probability of a type I error for the chosen region of part (a)? Does the region specify a level .05 test? Is it the best level .05 test?
c. If 60% of all enthusiasts prefer gut, calculate the probability of a type II error using the appropriate region from part (a). Repeat if 80% of all enthusiasts prefer gut.
d. If 13 out of the 20 players prefer gut, should $H _ { 0 }$ 
be rejected using a significance level of .10?

Each of a Group of 20 Intermediate Tennis Players Is H0:p=.5H _ { 0 } : p = .5H0​:p=.5

Each of a Group of 20 Intermediate Tennis Players Is $H _ { 0 } : p = .5$