Question

Chronic anterior compartment syndrome is a condition characterized by exercise-induced pain in the lower leg. Swelling and impaired nerve and muscle function also accompany this pain, which is relieved by rest. Susan Beckham and colleagues conducted an experiment involving 10 healthy runners and 10 healthy cyclists to determine whether there are significant differences in pressure measurements within the anterior muscle compartment for runners and cyclists. \(^{7}\) The data summary - compartment pressure in millimeters of mercury \((\mathrm{Hg})\) -is as follows: a. Test for a significant difference in the average compartment pressure between runners and cyclists under the resting condition. Use \(\alpha=.05 .\) b. Construct a \(95 \%\) confidence interval estimate of the difference in means for runners and cyclists under the condition of exercising at \(80 \%\) of maximal oxygen consumption. c. To test for a significant difference in the average compartment pressures at maximal oxygen consumption, should you use the pooled or unpooled \(t\) -test? Explain.

Short answer

Question: Conduct a two-sample t-test for the resting condition and provide the result. To answer this question, please provide the following information: 1. The sample mean, sample standard deviation, and sample size for both groups (runners and cyclists) under the resting condition. 2. The critical value for the two-tailed test with α = 0.05 and degrees of freedom calculated using the Welch-Satterthwaite equation. 3. The test statistic calculated using the given formula. Based on this information, I can help you with the two-sample t-test for the resting condition and provide the result.

Step-by-step solution

a. Two-sample t-test for the resting condition

To determine if there is a significant difference in the average compartment pressure between runners and cyclists under the resting condition, perform a two-sample t-test. 1. State the null and alternative hypotheses: \(H_0: \mu_{runners} = \mu_{cyclists}\) \(H_a: \mu_{runners} \neq \mu_{cyclists}\) 2. Use the given data to calculate the sample mean, sample standard deviation, and sample size for both groups (runners and cyclists) under the resting condition. 3. Calculate the test statistic: \(t = \frac{(\bar{x}_{runners} - \bar{x}_{cyclists}) - 0}{\sqrt{\frac{s_{runners}^2}{n_{runners}} + \frac{s_{cyclists}^2}{n_{cyclists}}}}\) 4. Find the critical value for the two-tailed test with \(\alpha = 0.05\) and degrees of freedom calculated using the Welch-Satterthwaite equation. 5. Compare the test statistic calculated in step 3 with the critical value obtained in step 4. If the test statistic lies outside the critical region, reject the null hypothesis and conclude that there is a significant difference in the average compartment pressure for runners and cyclists under the resting condition.

b. 95% Confidence Interval estimate

1. Calculate the standard error of the difference in means: \(SE = \sqrt{\frac{s_{runners}^2}{n_{runners}} + \frac{s_{cyclists}^2}{n_{cyclists}}}\) 2. Find the t-score corresponding to a 95% confidence level and the degrees of freedom calculated using the Welch-Satterthwaite equation. 3. Calculate the margin of error: \(ME = t_{score} * SE\) 4. Compute the confidence interval estimate for the difference in means: \(CI = (\bar{x}_{runners} - \bar{x}_{cyclists}) \pm ME\)

c. Pooled or Unpooled t-test

To decide whether to use the pooled or unpooled t-test, compare the sample variances or standard deviations of runners and cyclists under maximal oxygen consumption. 1. Calculate the ratio of the larger variance to the smaller variance. If this ratio is close to 1 (say, less than 2), then variances are similar, and a pooled t-test can be used. Otherwise, use the unpooled t-test. 2. Another approach is to perform an F-test for the equality of variances. If the p-value of the F-test is greater than a predetermined significance level (e.g. 0.10), we can assume that variances are equal and then use the pooled t-test. If the p-value is less than the significance level, use the unpooled t-test. By following these steps, you will be able to analyze the compartment pressure measurements for runners and cyclists and draw conclusions based on the given data.

Key concepts

Confidence Interval

A confidence interval provides a range of values that is likely to contain the true difference in means between two groups. In this exercise, it helps estimate the difference in compartment pressures for runners and cyclists during exercise at 80% of maximal oxygen consumption. Constructing a confidence interval involves several key steps.

• Calculate the standard error (SE) of the difference in means. This accounts for variations in sample data and is calculated as:\[SE = \sqrt{\frac{s_{runners}^2}{n_{runners}} + \frac{s_{cyclists}^2}{n_{cyclists}}}\]
• Find the appropriate t-score for a 95% confidence level using the calculated degrees of freedom through the Welch-Satterthwaite equation. This t-score reflects the reliability of our estimate.
• Determine the margin of error (ME) by multiplying the t-score by the SE:\[ME = t_{score} \times SE\]
• Finally, the confidence interval for the mean difference is given by:\[CI = (\bar{x}_{runners} - \bar{x}_{cyclists}) \pm ME\]This interval allows us to say that we are 95% confident that the true difference in mean pressures lies within this range.

Difference in Means

Evaluating the difference in means is crucial to understanding if two groups differ significantly when tested. Here, we compare the average compartment pressure between runners and cyclists to determine if there's a consistent difference. The two-sample t-test is an effective method to analyze this difference.

• The null hypothesis \(H_0\) states that there is no difference: \(\mu_{runners} = \mu_{cyclists}\). Conversely, the alternative hypothesis \(H_a\) suggests that a difference exists: \(\mu_{runners} eq \mu_{cyclists}\).
• To test these hypotheses, calculate the test statistic \(t\), using:\[t = \frac{(\bar{x}_{runners} - \bar{x}_{cyclists})}{\sqrt{\frac{s_{runners}^2}{n_{runners}} + \frac{s_{cyclists}^2}{n_{cyclists}}}}\]This statistic helps identify if observed differences are due to random sampling or actual variations.
• The t-test result is compared against a critical t-value for the specific significance level (\(\alpha = 0.05\)). If the result is extreme enough, it suggests a significant difference in pressures, prompting rejection of the null hypothesis.

Variance Comparison

Determining whether to use a pooled or unpooled t-test heavily relies on comparing the variances of the two groups. This step checks if the variances of compartment pressures for runners and cyclists are similar under maximal oxygen consumption.

• Calculate the ratio of larger to smaller variance. If this ratio is close to 1 (usually less than 2), the variances can be assumed similar, allowing for a pooled t-test.
• For a concrete decision, conduct an F-test, which compares the variances. If the p-value from this test is greater than a set significance level (such as 0.10), assume equal variances; otherwise, use the unpooled t-test.

Pooled t-tests offer simplicity when variances are equal, by combining variance estimates from both groups. Conversely, unpooled (or Welch's) t-tests handle discrepancies between variances, providing a more accurate analysis when variances differ significantly.