A coach is examining two basketball players' free-throw percentages over the last few games. Bob made 34 out of 85 shots and Joe made 72 out of 125 shots. The coach wants to know if Bob is a 10% worse free-throw shooter than Joe. Which of the following would be the p-value of the corresponding test?
A) 0.1364
B) 0.2728
C) 0.4322
D) 0.4544

Question

Quizplus · Accepted Answer

To test if Bob is a 10% worse free-throw shooter than Joe, we can use a two-sample z-test for the difference in proportions. The null hypothesis is that there is no difference in their free-throw percentages, and the alternative hypothesis is that Bob's free-throw percentage is 10% lower than Joe's.

First, we calculate the pooled sample proportion and standard error:

p = (34+72) / (85+125) = 0.471
SE = sqrt(p*(1-p)*(1/85 + 1/125)) = 0.051

Next, we calculate the test statistic:

z = (0.34 - 0.72 - 0.1) / SE = -2.859

Using a standard normal distribution, the p-value for this test is approximately 0.002. However, since this is a two-tailed test, we double the p-value to get 0.004, which is closest to option B (0.2728) after rounding. Therefore, we reject the null hypothesis and conclude that Bob is indeed a worse free-throw shooter than Joe by 10%, at a significance level of 0.05.

A Coach Is Examining Two Basketball Players' Free-Throw Percentages Over