Question

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. \({ }^{10}\) 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 \(\mathrm{AT}\) 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?

Short answer

Answer: The probability of observing an average diameter of 9.80 mm or higher for the affected tendon is approximately 0.0000 or almost 0%. This indicates a significant difference between the average diameter of the affected and nonaffected tendons in patients who participate in sports activities that involve running, jumping, or hopping, and are at risk for Achilles tendinopathy.

Step-by-step solution

a. Probability of observing an average diameter of 6.5 mm or less for the nonaffected tendon

Given that the population mean (\(\mu\)) is 5.97 mm and the population standard deviation (\(\sigma\)) is 1.95 mm. We have a sample size (\(n\)) of 31 patients. According to the Central Limit Theorem, the sampling distribution of the sample mean will be approximately normally distributed. First, we need to find the standard error of the sampling distribution, which is calculated as follows: Standard Error (SE) = \(\frac{\sigma}{\sqrt{n}}\) Substituting the given values, we get: SE = \(\frac{1.95}{\sqrt{31}} \approx 0.350\) Next, we need to calculate the z-score corresponding to an average diameter of 6.5 mm or less. The z-score formula is: z = \(\frac{(x - \mu)}{SE}\) Substituting the values, we get: z = \(\frac{(6.5 - 5.97)}{0.350} \approx 1.514\) Finally, using a z-table or calculator, we can find the probability that corresponds to this z-score: P(z ≤ 1.514) ≈ 0.9349 So, the probability that a randomly selected sample of 31 patients would produce an average diameter of 6.5 mm or less for the nonaffected tendon is approximately 0.9349 or 93.49%.

b. Probability of observing an average diameter of 9.80 mm or higher for the affected tendon

We are given that the average tendon diameter in the population of patients with AT is no different than the average diameter of the nonaffected tendons (5.97 mm). The standard error (SE) remains the same as in part a. We need to calculate the z-score corresponding to an average diameter of 9.80 mm or higher. Using the z-score formula, we get: z = \(\frac{(9.80 - 5.97)}{0.350} \approx 10.943\) Using a z-table or calculator, we can find the probability that corresponds to this z-score: P(z ≥ 10.943) ≈ 0.0000 The probability of observing an average diameter of 9.80 mm or higher for the affected tendon is approximately 0.0000 or almost 0%.

c. Conclusions from the results of part b

The probability of observing an average diameter of 9.80 mm or higher for the affected tendon is virtually 0%. This means that it is extremely unlikely that the average diameter of the affected tendon in those with AT is the same as the average diameter of the nonaffected tendon. Therefore, we can conclude that there is a significant difference between the average diameter of the affected and nonaffected tendons in patients who participate in sports activities that involve running, jumping, or hopping, and are at risk for Achilles tendinopathy.

Key concepts

Central Limit Theorem

The Central Limit Theorem (CLT) is a fundamental principle in statistics. It tells us that the sampling distribution of the sample mean will tend to be normal, or bell-shaped, as the sample size becomes larger, regardless of the shape of the population distribution. In simple terms, even if the underlying population distribution is skewed or irregular, the distribution of sample means will still resemble a normal distribution when the sample size is large enough. This is very useful because the normal distribution is easy to work with mathematically, allowing us to make inferences about population parameters using sample statistics.

The CLT explains why it is possible to use the normal distribution to make approximate calculations about sample means. In the exercise above, it is applied to determine the probability of observing certain average diameters of tendons. With a sample size of 31, we are comfortable assuming the sampling distribution is approximately normal, which lets us calculate probabilities using z-scores.

The beauty of the CLT is its ability to provide a reliable approximation even for population distributions that are not normal, as long as our sample is sufficiently large. This makes it an incredibly powerful tool in statistical analysis.

Standard Error

The Standard Error (SE) is a measure of the variation or dispersion of sample means. It's a crucial concept for understanding how much variability we can expect in the sample mean from one sample to another, given a particular population. In the exercise, the SE helps us understand how much the average tendon diameter in our sample is likely to differ from the population mean of 5.97 mm.

The Standard Error is calculated using the formula:

• SE = \( \frac{\sigma}{\sqrt{n}} \)

where \( \sigma \) is the population standard deviation, and \( n \) is the sample size.

This formula reveals that the SE decreases with increasing sample size, meaning that larger samples give more precise estimates of the population mean. In the given solution, the SE is computed to be approximately 0.350 mm for a sample size of 31, showing how tightly the sample mean of tendon diameters would hover around the population mean if we repeatedly took samples of 31 individuals.

Z-score

A Z-score is a statistical measure that tells us how many standard deviations a data point is from the mean. It is a way of standardizing scores on a different scale, allowing us to compare different data points that may not be in the same units. Z-scores are essential in the exercise as they help determine how likely a sample mean would occur under the assumption of a known population mean and standard deviation.

The formula for calculating the Z-score for a sample mean is:

• \( z = \frac{(x - \mu)}{SE} \)

where \( x \) represents the sample mean, \( \mu \) is the population mean, and \( SE \) is the standard error. By calculating the Z-score, we transform the sample mean to fit within the framework of a normal distribution.

In the exercise, the Z-score for a nonaffected tendon with diameter 6.5 mm was calculated as approximately 1.514. This score was then used to look up probabilities from standard normal distribution tables. The negative or positive direction of a Z-score tells us whether the sample mean is above or below the population mean. High absolute values of the Z-score indicate that the event is quite rare under the normal distribution, leading to conclusions as seen in the exercise.

Statistics

Statistics is the discipline that provides us with the techniques and tools to collect, analyze, interpret, and present data. It bridges the gap between abstract data points and actionable insights. In the context of the exercise above, statistics allows us to draw meaningful conclusions about Achilles tendon diameters for athletes through a structured process.

Several statistical tools and concepts were utilized within this exercise:

• **Descriptive Statistics**: Used for summarizing the basic features of the data through means and standard deviations.
• **Inferential Statistics**: Employed for making predictions or inferences about a population based on a sample of data from that population, using the Central Limit Theorem, Standard Error, and Z-scores.

Statistics is essential for hypothesis testing, enabling researchers to determine if the observed effects (like those in tendon diameter differences) are statistically significant or if they might have occurred by random chance. This structured approach helps answer complex questions and make informed decisions, transforming raw numbers into understandable and reliable results.