Forty-five members of a bowling league sign up for a program that claims to improve bowling scores. The participants bowl a set of five games before the program begins and a set of five games again at the end to measure their improvement. The mean number of points improved (over the set of five games) was $\bar { x } = 18$. Assume the standard deviation is $\sigma = 59$ and let $\mu$ be the population mean number of points improved for the set of five games. To determine whether the program is effective, a test is made of the hypotheses $H _ { 0 } : \mu = 0$ versus $H _ { 1 } : \mu 0$.
Do you reject $H _ { 0 }$ at the $\alpha = 0.01$ level?
A) No
B) Yes
C) There is not enough information to draw a conclusion.

Question

Forty-five members of a bowling league sign up for a program that claims to improve bowling scores. The participants bowl a set of five games before the program begins and a set of five games again at the end to measure their improvement. The mean number of points improved (over the set of five games) was $\bar { x } = 18$. Assume the standard deviation is $\sigma = 59$ and let $\mu$ be the population mean number of points improved for the set of five games. To determine whether the program is effective, a test is made of the hypotheses $H _ { 0 } : \mu = 0$ versus $H _ { 1 } : \mu > 0$.
Do you reject $H _ { 0 }$ at the $\alpha = 0.01$ level?
A) No
B) Yes
C) There is not enough information to draw a conclusion.

Quizplus · Accepted Answer

To determine whether to reject $H_0$, we calculate the z-score using the formula $z = \frac{\bar{x} - \mu}{\sigma / \sqrt{n}}$, where $\bar{x} = 18$, $\mu = 0$, $\sigma = 59$, and $n = 45$. Plugging in these values gives $z = \frac{18 - 0}{59 / \sqrt{45}}$, which results in a z-score that we can compare against the critical value for $\alpha = 0.01$. Given the standard normal distribution, the critical z-value for a one-tailed test at $\alpha = 0.01$ is approximately 2.33. If the calculated z-score is less than 2.33, we do not reject $H_0$. Without calculating the exact z-score here, the process described is how one would determine whether to reject $H_0$ or not, and the question implies that the evidence is not strong enough to reject $H_0$ at the $\alpha = 0.01$ level, assuming standard procedures and typical statistical thresholds are followed.

Forty-Five Members of a Bowling League Sign Up for a Program