Forty 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 $\bar { x } = 11$. Assume the standard deviation is $\sigma = 49$ 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) Yes
B) There is not enough information to draw a conclusion.
C) No

Question

Forty 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 $\bar { x } = 11$. Assume the standard deviation is $\sigma = 49$ 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) Yes
B) There is not enough information to draw a conclusion.
C) No

Quizplus · Accepted Answer

We need to perform a one-tailed t-test since the alternative hypothesis is stating an increase in the mean score. The formula for the t-test is:

$$t=\frac{\bar{x}-\mu}{s/\sqrt{n}}$$

Where $\bar{x}$ is the sample mean, $\mu$ is the population mean, $s$ is the sample standard deviation, and $n$ is the sample size.

Plugging in the given values, we get:

$$t=\frac{11-0}{49/\sqrt{40}}=3.09$$

With 39 degrees of freedom (n-1). Using a t-distribution table with a significance level of 0.01 and 39 degrees of freedom, the critical value for a one-tailed test is 2.423.

Since our calculated t-value (3.09) is greater than the critical value (2.423), we reject the null hypothesis. Therefore, we can conclude that the program is effective in improving bowling scores.

Forty Members of a Bowling League Sign Up for a Program