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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.

Step by step solution

01

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

02

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.

03

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.

04

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.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Standard Error

The standard error (SE) is a measure that lets us determine how close the sample mean is likely to be to the population mean. It essentially tells us how much sampling variability is present.
The concept of standard error is crucial in calculating confidence intervals, as it gauges the precision of the sample mean.
In our baseball exercise context, the SE is specifically used to estimate the standard error of the difference between mean hip range of motions for two independent groups: pitchers and position players. With the given data:

• Sample size for pitchers ( n1) = 40
• Sample size for position players ( n2) = 40
• Standard deviation for pitchers ( s1) = 5.9
• Standard deviation for position players ( s2) = 7.6

The formula used for SE is:\[SE = \sqrt{\frac{(s1)^2}{n1} + \frac{(s2)^2}{n2}}\]By substituting the values, you can compute the SE, which in turn, is essential for calculating the confidence interval of the mean difference in hip motion.

Sample Mean

The sample mean is a fundamental concept in statistics that represents the average of a set of data points collected from a sample. In simple terms, it provides a central value. It is essential for estimating the population mean, which is the true mean that we actually want to understand. In the context of our exercise, we have two separate sample means:

• For pitchers, the sample mean ( x1) is 75.6 degrees.
• For position players, the sample mean ( x2) is 79.6 degrees.

The goal is to find out how these means differ by calculating a confidence interval.
Ultimately, the sample means serve as the basis for calculating the predicted average hip range of motion for the entire population of pitchers and position players. This helps in comparing whether there is indeed a difference in movement between the two groups.

Standard Deviation

Standard deviation is a measure that indicates the extent of deviation for a group as a whole. A lower standard deviation means that the data points are close to the mean, while a higher standard deviation means that the data points are more spread out from the mean.
In our baseball problem, standard deviation helps us understand the variability in hip range of motion among players:

• Pitchers have a standard deviation ( s1) of 5.9 degrees.
• Position players have a standard deviation ( s2) of 7.6 degrees.

Standard deviation is instrumental in the calculation of the standard error, which further plays a role in deriving the confidence interval of the mean difference.
Understanding variability within each group provides insights into the consistency of the players' movement performances.

Independent Samples

Independent samples refer to two or more groups of responses that are collected from different entities without any influence on each other. The samples have no connection, ensuring they do not affect one another.
In our context, the baseball player's study involves two independent samples: pitchers and position players. Each group provides data regarding their respective hip range of motion.
The independence ensures that the results from one group don't interfere with the results from the other, allowing for fair comparison.
When creating confidence intervals, it is crucial to use independent samples to ensure that the results are statistically valid and reliable. This concept ensures accuracy when comparing the true mean differences between the two groups.