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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

Identify the given values

First, let's list down the given values from the problem: For pitchers (group 1): - Sample size (n1): 40 - Mean (M1): 75.6 degrees - Standard deviation (SD1): 5.9 degrees For position players (group 2): - Sample size (n2): 40 - Mean (M2): 79.6 degrees - Standard deviation (SD2): 7.6 degrees Confidence level: 90%

02

Calculate the standard error of the difference

We'll calculate the standard error of the difference between the means using the following formula: \[SE_{diff} = \sqrt{\frac{SD1^2}{n1} + \frac{SD2^2}{n2}}\] Plugging in the given values: \(SE_{diff} = \sqrt{\frac{5.9^2}{40} + \frac{7.6^2}{40}}\) Calculate the standard error of the difference: \(SE_{diff} ≈ 1.42\)

03

Find the critical value for the 90% confidence interval

To find the critical value for a 90% confidence interval, we can look up the value in a Z-table for a standard normal distribution: For 90% confidence, in a two-tailed test, α is split into two, which mean α/2 = 0.05. Looking up this value in the Z-table gives a critical value of approximately 1.645.

04

Calculate the margin of error

The margin of error is calculated by multiplying the critical value by the standard error of the difference: Margin of error = Critical value (Z-value) × Standard error of the difference Margin of error = 1.645 × 1.42 ≈ 2.34

05

Compute the confidence interval

Now, we can compute the 90% confidence interval of the difference in mean hip range of motion for pitchers and position players: - Lower limit: (M1 - M2) - Margin of error - Upper limit: (M1 - M2) + Margin of error Plug in the values: - Lower limit: (75.6 - 79.6) - 2.34 ≈ -6.34 - Upper limit: (75.6 - 79.6) + 2.34 ≈ -2.66 The 90% confidence interval for the difference in mean hip range of motion between pitchers and position players is approximately (-6.34, -2.66) degrees. This means we can be 90% confident that the true difference in mean hip range of motion between pitchers and position players lies within this range.

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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 method used in statistics to determine if there is enough evidence to reject a null hypothesis. In the context of the exercise, our null hypothesis might state that there is no difference in hip range of motion between pitchers and position players. To test this, we compare the means of two independent samples—pitchers and position players.

• Null Hypothesis (H_0): The mean hip range of motion for pitchers is equal to that of position players (\(\mu_1 = \mu_2\house, even if the results seem significant, there's always a small chance of rejecting the null hypothesis when it is actually true (type I error\): \).
• Alternative Hypothesis (H_a): The means are not equal (\(\mu_1 eq \mu_2\about 5\ this level (usually \lconsis \ house \texcure\
hip\

Testing the hypothesis helps in deciding whether the observed data Supports the claim of a difference in means. The hypothesis is either confirmed, or the null hypothesis is rejected, providing statistical evidence for findings.

Sampling Distribution

The sampling distribution is the distribution of a particular statistic, like the sample mean, across different possible samples from the same population. Here's why it matters in hypothesis testing and constructing confidence intervals:

• Sample Mean: Each sample from a population has its own mean, and the collection of these means forms the sampling distribution. For instance, each sample of pitchers or position players has an average hip range of motion, and these averages are part of their respective sampling distributions.
• Central Limit Theorem: The Central Limit Theorem lets us know that the sampling distribution of the sample mean will be approximately normally distributed. This property holds true as long as the sample size is large enough (usually ≥ 30). In this case, both samples (n=40) satisfy this condition.

Understanding the sampling distribution is key to determining how sample statistics relate to population parameters. It also helps in estimating the accuracy of sample mean as a representation of the population mean.

Standard Error

The standard error is a critical component in confidence interval estimation and hypothesis testing.It measures the variability of sample means around the population mean. It helps assess the reliability of sample statistics:

• Calculation: For the difference between two sample means, the standard error (SE_{diff}) is calculated using the formula:\[SE_{diff} = \sqrt{\frac{SD1^2}{n1} + \frac{SD2^2}{n2}}\]This formula takes into account the standard deviation and size of each sample.
• Interpretation: A smaller standard error indicates more precise estimates of the population mean differences. In our exercise, the calculated standard error of approximately 1.42 suggests a relatively precise estimate for the mean difference in hip range of motion.

The importance of standard error lies in its role in determining how much the sample mean could fluctuate due to random sampling variability. This fluctuation affects the conclusions drawn from any statistical analysis.

Margin of Error

The margin of error represents the range within which we expect the true population parameter to lie.It accounts for the sampling error in our estimate of the population mean:

• Calculation: The margin of error is determined by multiplying the standard error by the critical value from a Z-distribution table:\[Margin\ of\ Error = Critical\ Value \times SE_{diff}\]For a 90% confidence interval, the critical value is around 1.645.
• Application: In the exercise, multiplying the standard error by 1.645 gives a margin of error of approximately 2.34. This value tells us that we can be 90% certain that the true mean difference is within ±2.34 degrees of the observed difference.

Including the margin of error in a confidence interval helps in assessing how much the observed sample mean allows us to generalize for the population. It essentially quantifies the uncertainty around this estimation.