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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 of independent samples of 40 professionul 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, whercas 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 bascball pitchers and position players, test hypotheses appropriate for determining if there is convincing evidence that the mean range of motion for pitchers is less than the mean for position players.

Step by step solution

01

Organize the data

Before conducting the test, let's first organize the available data. - Sample size for pitchers: \(n_p = 40\) - Sample mean hip range of motion for pitchers: \(\bar{x}_p = 75.6\) degrees - Sample standard deviation for pitchers: \(s_p = 5.9\) degrees - Sample size for position players: \(n_{pp} = 40\) - Sample mean hip range of motion for position players: \(\bar{x}_{pp} = 79.6\) degrees - Sample standard deviation for position players: \(s_{pp} = 7.6\) degrees

02

Compute test statistic

To perform the two-sample t-test for independent means, we need to calculate the t-value. The equation for the t-value is \[t = \frac{(\bar{x}_p - \bar{x}_{pp}) - (\mu_p - \mu_{pp})}{\sqrt{\frac{s_p^2}{n_p} + \frac{s_{pp}^2}{n_{pp}}}}\] Assuming the null hypothesis to be true, the means are equal so \(\mu_p - \mu_{pp} = 0\). The t-value then simplifies to \[t = \frac{(\bar{x}_p - \bar{x}_{pp})}{\sqrt{\frac{s_p^2}{n_p} + \frac{s_{pp}^2}{n_{pp}}}}\] Now, let's plug in the values: \[t = \frac{(75.6 - 79.6)}{\sqrt{\frac{5.9^2}{40} + \frac{7.6^2}{40}}}\]

03

Calculate the t-value

After substituting the values in the t-value formula, we can perform the calculations: \[t = \frac{(-4)}{\sqrt{\frac{34.81}{40} + \frac{57.76}{40}}} = \frac{-4}{\sqrt{0.87025 + 1.444}} = -3.33\]

04

Compute the degrees of freedom

Next, to determine the p-value, we need to compute the degrees of freedom (df) using the following equation: \[df \approx \frac{(\frac{s_p^2}{n_p} + \frac{s_{pp}^2}{n_{pp}})^2}{\frac{(\frac{s_p^2}{n_p})^2}{n_p - 1} + \frac{(\frac{s_{pp}^2}{n_{pp}})^2}{n_{pp} - 1}}\] Plugging in the known values: \[df \approx \frac{(0.87025 + 1.444)^2}{\frac{(0.87025)^2}{39} + \frac{(1.444)^2}{39}}\]

05

Calculate the degrees of freedom

Now, we calculate the degrees of freedom: \[df \approx \frac{(2.31425)^2}{\frac{(0.87025)^2}{39} + \frac{(1.444)^2}{39}} = 72.04\] Rounded down to the nearest whole number, we get df = 72.

06

Calculate the p-value

With a t-value of -3.33 and df = 72, we need to calculate the p-value using a two-sample t-distribution table or a calculator. In this case, we can estimate the p-value: \(p \approx 0.001\). Remember that we are testing the hypothesis with a one-tailed test where the mean range of motion for pitchers is less than the mean for position players (\(\mu_p < \mu_{pp}\)).

07

Make a decision

Now it's time to make a decision regarding the hypothesis by comparing the p-value to a predefined significance level (\(\alpha\)). If the p-value is less than \(\alpha = 0.05\), we reject the null hypothesis in favor of the alternative hypothesis. Since the p-value (\(p \approx 0.001\)) is less than the significance level (\(\alpha = 0.05\)), we reject the null hypothesis. #Conclusion#: There is convincing evidence that the mean range of motion for professional baseball pitchers is less than the mean range of motion for position players.

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Key Concepts

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

Hypothesis Testing

Hypothesis testing is a fundamental statistical method used to determine if there is enough evidence to reject a null hypothesis, which is a baseline presumption, in favor of an alternative hypothesis. In the context of the given exercise involving baseball players, researchers aim to examine whether the sample data indicate a significant difference in hip range of motion between pitchers and position players. To do so, they set up two opposing hypotheses: a null hypothesis that assumes no difference in the populations' mean hip ranges of motion and an alternative hypothesis suggesting pitchers have a lower mean than position players. The outcome of hypothesis testing is not just to choose one hypothesis over the other but to measure the strength of the evidence against the null hypothesis.

Degrees of Freedom

Degrees of freedom (df) in statistics refers to the number of values in a calculation that are free to vary. When conducting the two-sample t-test as seen in the exercise, the degrees of freedom are particularly important for determining the critical value or p-value from the t-distribution. The df for a two-sample t-test is calculated based on the sample sizes and their variances, which is used to reference the correct distribution for determining statistical significance. Generally, as df increases, the t-distribution approaches the normal distribution.

Sample Mean

The sample mean is the average value of a sample, which in the exercise is represented by the mean hip range of motion for the two groups of baseball players. It is calculated by summing all the values and dividing by the number of observations. In hypothesis testing, the sample mean is a crucial statistic, typically denoted as \(\bar{x}\), and it is used to estimate the population mean, informing our judgment about the null hypothesis.

Sample Standard Deviation

Sample standard deviation quantifies the amount of variation within a set of sample measurements. It tells us how spread out the numbers in our sample are in relation to the sample mean. In hypothesis testing, like the two-sample t-test conducted in the exercise, the sample standard deviation is used to help calculate the test statistic, which measures the difference between the sample means relative to the variability within the samples. A high standard deviation indicates that the sample points can vary widely, which can affect the reliability of the comparison.

P-Value

The p-value is a measure that helps us determine the strength of the results in hypothesis testing. It's the probability of observing the test statistic or one more extreme, assuming that the null hypothesis is true. In simple terms, a small p-value indicates that what we're observing is unlikely to have occurred by random chance if there were no actual difference (or effect). In the exercise, a p-value of approximately 0.001 suggests very strong evidence against the null hypothesis, as it is much lower than the common significance level of 0.05.

Null Hypothesis

The null hypothesis (\(H_0\)) is a statement in hypothesis testing that there is no effect or no difference between groups. It essentially posits the 'status quo' and acts as the assumption to be challenged by the alternative hypothesis. In the exercise with baseball players, the null hypothesis would state that there is no difference in the mean hip range of motion between pitchers and position players.

Alternative Hypothesis

In contrast to the null hypothesis, the alternative hypothesis (\(H_a\)) represents the statement that there is an effect or a difference that the study aims to detect. It’s what we hope to conclude if the evidence against the null hypothesis is strong enough. In our exercise, the alternative hypothesis suggests that pitchers have a lower mean hip range of motion compared to position players. Choosing whether the alternative hypothesis is one-sided (as in this case) or two-sided depends on the specific question the research is trying to answer.

Statistical Significance

Statistical significance indicates whether the results obtained from a study or an experiment are likely to be meaningful or just caused by chance. It's often determined by using a significance level (denoted as \(\alpha\)), which sets the threshold for how extreme the results must be to reject the null hypothesis. As observed in the exercise, the p-value was substantially less than the typical significance level of 0.05 (5%), leading to the conclusion that there is a statistically significant difference in the mean hip range of motion between the two types of baseball players.