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Some sports that involve a significant amount of running, jumping, or hopping put participants at risk for Achilles tendinopathy (AT), an inflammation and thickening of the Achilles tendon. A study in The American Journal of Sports Medicine looked at the diameter (in \(\mathrm{mm}\) ) of the affected and nonaffected tendons for patients who participated in these types of sports activities. \(^{9}\) Suppose that the Achilles tendon diameters in the general population have a mean of 5.97 millimeters (mm) with a standard deviation of \(1.95 \mathrm{~mm} .\) a. What is the probability that a randomly selected sample of 31 patients would produce an average diameter of \(6.5 \mathrm{~mm}\) or less for the nonaffected tendon? b. When the diameters of the affected tendon were measured for a sample of 31 patients, the average diameter was \(9.80 .\) If the average tendon diameter in the population of patients with AT is no different than the average diameter of the nonaffected tendons \((5.97 \mathrm{~mm})\), what is the probability of observing an average diameter of 9.80 or higher? c. What conclusions might you draw from the results of part b?

Step by step solution

01

Write the formulas for the sampling distribution and z-score

We'll use the follow formulas: 1. Mean of the sampling distribution of the sample mean, \(\mu_{\bar{x}}=\mu\) 2. Standard deviation of the sampling distribution, \(\sigma_{\bar{x}}=\frac{\sigma}{\sqrt{n}}\) 3. Z-score: \(z=\frac{\bar{x}-\mu_{\bar{x}}}{\sigma_{\bar{x}}}\)

02

Calculate the standard deviation of the sampling distribution for nonaffected tendons

Given \(\sigma=1.95\) mm and \(n=31\), we can calculate \(\sigma_{\bar{x}}\). \(\sigma_{\bar{x}}=\frac{1.95}{\sqrt{31}} = 0.35\) mm

03

Calculate the z-score for a sample mean of 6.5 mm for nonaffected tendons

Given \(\mu=5.97\) mm and \(\bar{x}=6.5\) mm, we can calculate the z-score. \(z=\frac{6.5-5.97}{0.35}=1.51\)

04

Find the probability for a sample mean of 6.5 mm or less for nonaffected tendons

From a z-table or using a calculator, we find the probability corresponding to a z-score of 1.51. \(P(Z \le 1.51) = 0.9347\) The probability of a sample mean of 6.5 mm or less for nonaffected tendons is 0.9347.

05

Calculate the z-score for a sample mean of 9.80 mm for affected tendons

Using the same formulas from steps 2 and 3, but with the assumption that the population mean for affected tendons is the same as nonaffected tendons, we can find the z-score for an average diameter of 9.80 mm in affected tendons. \(z=\frac{9.80-5.97}{0.35}=10.94\)

06

Find the probability for a sample mean of 9.80 mm or higher for affected tendons

We will find the probability associated with the z-score of 10.94. However, since the z-score is extremely high, the probability will be very close to 0. It is virtually impossible to observe an average diameter of 9.80 mm or higher for affected tendons if their mean is the same as nonaffected tendons.

07

Draw conclusions

Based on the result from part b, the probability of observing a sample mean of 9.80 mm or higher for affected tendons if their mean was the same as nonaffected tendons is almost zero. Therefore, we can conclude that the average tendon diameter of affected tendons is significantly larger than the average diameter of nonaffected tendons. This is consistent with the fact that affected tendons have inflammation and thickening.

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Key Concepts

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

Sampling Distribution

The concept of a sampling distribution is central to understanding how we make inferences about populations. A sampling distribution refers to the probability distribution of a given statistic based on a random sample. It helps us know how the sample mean would behave if we took many samples from the population.

• The mean of the sampling distribution, denoted as \( \mu_{\bar{x}} \), is the same as the population mean \( \mu \).
• The standard deviation, \( \sigma_{\bar{x}} \), is computed as \( \frac{\sigma}{\sqrt{n}} \), where \( n \) is the sample size and \( \sigma \) is the population standard deviation.

In this exercise, knowing about sampling distributions helps us estimate probabilities regarding Achilles tendon diameters in both affected and unaffected groups.

Z-score

A Z-score is a numerical measurement that describes a value's relationship to the mean of a group of values. It is expressed in terms of standard deviations from the mean. A Z-score of 0 indicates that the data point's score is identical to the mean score.

• To calculate a Z-score, use the formula \( z=\frac{\bar{x}-\mu_{\bar{x}}}{\sigma_{\bar{x}}} \).
• A positive Z-score indicates that the data point is above the mean, while a negative score indicates it is below the mean.

In this situation, Z-scores help us determine the probability of observing certain mean diameters in tendon measurements by standardizing the values.

Standard Deviation

Standard deviation is a measure of the amount of variation or dispersion in a set of values. A low standard deviation indicates that the values tend to be close to the mean, while a high standard deviation indicates they are spread out over a wider range.

• In this exercise, the population standard deviation \( \sigma \) is given as 1.95 mm for nonaffected tendons.
• The standard deviation for the sampling distribution, \( \sigma_{\bar{x}} \), is calculated as \( 0.35 \) mm, which indicates how much the sample mean would vary from sample to sample.

Understanding standard deviation is crucial to interpret how typical the sampled data is compared to the population.

Achilles Tendinopathy

Achilles tendinopathy is a condition characterized by inflammation and thickening of the Achilles tendon. It is common in sports involving running, jumping, or hopping.

• This study investigates whether affected tendons have a significantly larger diameter compared to unaffected tendons.
• The findings showed that the probability of finding an average diameter of 9.80 mm or higher in affected tendons is virtually zero, assuming the same population mean.

This analysis supports medical observations that affected tendons are indeed thicker due to inflammation, aligning with the understanding of this medical condition.