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Research has shown that good hip range of motion and strength in throwing athletes results in improved performance and decreased body stress. The article "Functional Hip Characteristics of Baseball Pitchers and Position Players" (Am. J. Sport. Med., 2010: 383-388) reported on a study involving samples of 40 professional pitchers and 40 professional position players. For the pitchers, the sample mean trail leg total arc of motion (degrees) was \(75.6\) with a sample standard deviation of \(5.9\), whereas the sample mean and sample standard deviation for position players were \(79.6\) and 7.6, respectively. Assuming normality, test appropriate hypotheses to decide whether true average range of motion for the pitchers is less than that for the position players (as hypothesized by the investigators). In reaching your conclusion, what type of error might you have committed?

Step by step solution

01

Define the Hypotheses

We need to determine whether the average range of motion for pitchers is less than that for position players. Therefore, we set up the null and alternative hypotheses:- Null Hypothesis \( H_0: \mu_1 = \mu_2 \) (The true mean range of motion for pitchers \( \mu_1 \) is equal to that of position players \( \mu_2 \)).- Alternative Hypothesis \( H_a: \mu_1 < \mu_2 \) (The true mean range of motion for pitchers \( \mu_1 \) is less than that of position players \( \mu_2 \)).

02

Identify the Test Statistic

Since we are comparing means with known sample standard deviations, we use the two-sample t-test for the difference between means.The test statistic formula is:\[t = \frac{(\bar{x}_1 - \bar{x}_2)}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}}\]where \(\bar{x}_1 = 75.6\), \(s_1 = 5.9\), \(n_1 = 40\) for pitchers, and \(\bar{x}_2 = 79.6\), \(s_2 = 7.6\), \(n_2 = 40\) for position players.

03

Calculate the Test Statistic

Plug the given values into the test statistic formula:\[t = \frac{(75.6 - 79.6)}{\sqrt{\frac{5.9^2}{40} + \frac{7.6^2}{40}}} = \frac{-4.0}{\sqrt{0.86825}} = \frac{-4.0}{0.93161} \approx -4.29\]

04

Determine the Critical Value and Decision Rule

Using a significance level (\( \alpha \)) of 0.05, since the alternative hypothesis is a one-tailed test, we look up the critical value for a t-distribution with \(df \approx 78\) (using the larger of \(n-1\)) in a t-table. The critical value for a one-tailed test at \( \alpha = 0.05 \) is approximately \(-1.664\).Decision Rule: Reject \(H_0\) if the test statistic \(t < -1.664\).

05

Make a Decision

Since the calculated test statistic \(-4.29\) is less than the critical value \(-1.664\), we reject the null hypothesis. This suggests that the true average range of motion for pitchers is less than that of position players.

06

Consider Type of Error

Given that we rejected the null hypothesis, the type of error we might have committed is a Type I error. A Type I error occurs when the null hypothesis is true, but we incorrectly reject it.

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Key Concepts

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

Two-sample t-test

The two-sample t-test is a statistical method used to determine if there are significant differences between the means of two groups. In our exercise, we seek to compare the mean range of motion of pitchers and position players. This scenario is perfect for a two-sample t-test, as we are dealing with two independent groups where the sample standard deviations are known.

The formula for the two-sample t-test is:

• \[t = \frac{(\bar{x}_1 - \bar{x}_2)}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}} \]

Where:

• \( \bar{x}_1 \) and \( \bar{x}_2 \) are the sample means of the two groups.
• \( s_1 \) and \( s_2 \) are the sample standard deviations.
• \( n_1 \) and \( n_2 \) are the sample sizes.

The test statistic, \( t \), helps determine if the difference between the groups is large enough to be statistically significant. A calculated \( t \) value is compared to a critical \( t \) value from the t-distribution table. If it exceeds (or falls below for a one-tailed test) this threshold, we reject the null hypothesis.

Type I error

A Type I error occurs when we reject a true null hypothesis. This means that in our test, we concluded that the pitchers have a lower range of motion than position players when, in reality, there is no difference.

This error is often referred to as a "false positive," indicating that we observed a difference that doesn't actually exist. In hypothesis testing, the probability of committing a Type I error is denoted by the significance level \( \alpha \). By setting \( \alpha = 0.05 \), we aim to ensure that there is only a 5% chance of incorrectly rejecting the null hypothesis.

• Key factors influencing Type I error include:
• Choice of significance level (\( \alpha \)) - The lower the \( \alpha \), the lower the chance of a Type I error.
• Sample size - Larger samples can provide more reliable results, reducing the potential for errors.

It's important to balance Type I errors with Type II errors, where a false null hypothesis is not rejected.

Null and alternative hypotheses

In hypothesis testing, we use null and alternative hypotheses to assess a claim. For our exercise:

• **Null Hypothesis (\( H_0: \mu_1 = \mu_2 \))**: Assumes no difference in mean range of motion between pitchers and position players.
• **Alternative Hypothesis (\( H_a: \mu_1 < \mu_2 \))**: Claims that pitchers have less range of motion compared to position players.

Hypotheses are foundational to setting up the test. The null hypothesis is like the default assumption - believing there's no effect until proven otherwise. The alternative hypothesis is what researchers hope to support with their data.

These hypotheses are compared with the test statistic to evaluate their plausibility. If the test leads to a rejection of the null, we support the alternative. But remember, this doesn't "prove" the alternative, it merely provides evidence that is consistent with it.

Normal distribution assumption

The normal distribution assumption is crucial in conducting a valid two-sample t-test. This assumption suggests that the data in each group are approximately normally distributed. While real-world data may not perfectly adhere to normality, the Central Limit Theorem states that for samples larger than 30, the sample mean distribution tends to be normal.

In our scenario, both the pitchers and position players have 40 samples, which indicates that the t-test can proceed under the assumption of normality. However, realizing the importance of this assumption involves understanding why it's used:

• Ensures that the t-statistic follows the t-distribution, allowing for accurate decision-making.
• Justifies using the t-table to find critical values.

If the sample size is smaller or you suspect non-normality, consider graphical methods like Q-Q plots or statistical tests like the Shapiro-Wilk to check for normality. Otherwise, robust statistical methods that don't assume normality may be necessary.