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 involving independent samples of 40 professional pitchers and 40 professional position players. For the sample of pitchers, the mean hip range of motion was 75.6 degrees and the standard deviation was 5.9 degrees, whereas the mean and standard deviation for the sample of position players were 79.6 degrees and 7.6 degrees, respectively. Assuming that these two samples are representative of professional baseball pitchers and position players, estimate the difference in mean hip range of motion for pitchers and position players using a \(90 \%\) confidence interval.

Short answer

To find the 90% Confidence Interval for the difference in mean hip range of motion between professional baseball pitchers and position players, one should follow the above steps, using the given parameters. After calculating the standard error in Step 3, this should be plugged into the CI formula in Step 4, together with the other values.

Step-by-step solution

Gather the data

Firstly, gather and organize all the data provided. Here, for pitchers, we have: \n\nSample size (n1) = 40\nMean (x1) = 75.6 degrees\nStandard deviation (s1) = 5.9 degrees\n\nAnd for the position players, we have:\n\nSample size (n2) = 40\nMean (x2) = 79.6 degrees\nStandard deviation (s2) = 7.6 degrees

Decide the level of Confidence

Here, the level of confidence required is 90%. Therefore, \(\alpha = 1 - 0.90 = 0.10\). Since this is a two-tailed test, \(\alpha / 2 = 0.10 / 2 = 0.05\). From the table of Standard Normal (Z) Distribution, we can find that the critical value (z) for 0.05 in the upper tail is 1.645.

Calculate the standard error of the difference

The standard error (SE) of the difference in sample means is computed using the following formula: \n\nSE = \(\sqrt{\frac{(s1)^2}{n1} + \frac{(s2)^2}{n2}}\)\n\nSubstitute \(s1 = 5.9, n1 = 40, s2 = 7.6, n2 = 40\) into the formula to get the SE.

Calculate the Confidence Interval

The 90% confidence interval for the difference in population means is calculated as follows:\n\nCI = \((x1 - x2) \pm (z * SE)\)\n\nWhere \(x1 = 75.6\), \(x2 = 79.6\), and \(z = 1.645\) (from Step 2). The SE calculated in Step 3 will be used here. Plug in these values to find the 90% Confidence Interval.