Question

Research has shown that, for baseball players, good hip range of motion results in improved performance and decreased body stress. The article "Functional Hip Characteristics of Baseball Pitchers and Position Players" (The American Journal of Sports Medicine, \(2010: 383-388\) ) reported on a study of independent samples of 40 professional pitchers and 40 professional position players. For the pitchers, the sample mean hip range of motion was 75.6 degrees and the sample standard deviation was 5.9 degrees, whereas the sample mean and sample standard deviation for position players were 79.6 degrees and 7.6 degrees, respectively. Assuming that the two samples are representative of professional baseball pitchers and position players, test hypotheses appropriate for determining if mean range of motion for pitchers is less than the mean for position players.

Short answer

After calculating and comparing the test statistic and critical value, we can make a decision whether to reject or not reject the null hypothesis. This will indicate whether there is a statistically significant difference in mean hip range of motion between pitchers and position players.

Step-by-step solution

State the Hypotheses

The null hypothesis (\(H_0\)) and the alternative hypothesis (\(H_a\)) are defined as follows: \(H_0\): \(\mu_1 - \mu_2 = 0\) (There is no difference in mean range of motion between pitchers and position players). \(H_a\): \(\mu_1 - \mu_2 < 0\) (The mean range of motion for pitchers is less than that for position players). Where, \(\mu_1\) and \(\mu_2\) represent the population means for pitchers and position players, respectively.

Compute the Test Statistic

The test statistic for two independent samples is calculated as: \( z = \frac {(\bar{x}_1 - \bar{x}_2) - (\mu_1 - \mu_2)}{\sqrt{\frac{{s_1}^2}{n_1} + \frac{{s_2}^2}{n_2}}}\) Substituting the given values: \( z = \frac {(75.6 - 79.6) - 0}{\sqrt{\frac{{5.9}^2}{40} + \frac{{7.6}^2}{40}}}\) After calculation, we get the z-score.

Determine the Critical Value

Assuming a 0.05 level of significance (typical in social sciences), and because we stated a one-tailed (less than) hypothesis, we refer to the standard normal (Z) distribution table and find that the critical value is -1.645.

Make a Decision

We compare the calculated test statistic with the critical value. If the test statistic is less than the critical value, we reject the null hypothesis; otherwise, we do not reject it. This will show whether the mean hip range of motion for pitchers is statistically significantly less than that for position players.