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Chapter 10: Problem 27

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

Expert verified
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 伪 = 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

01

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.
02

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: \(CI = (\bar{x}_{runners} - \bar{x}_{cyclists}) \pm ME\)
03

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.

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Key Concepts

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

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.

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Most popular questions from this chapter

An experiment was conducted to compare the densities (in ounces per cubic inch) of cakes prepared from two different cake mixes. Six cake pans were filled with batter \(\mathrm{A}\), and six were filled with batter B. Expecting a variation in oven temperature, the experimenter placed a pan filled with batter \(\mathrm{A}\) and another with batter \(\mathrm{B}\) side by side at six different locations in the oven. The six paired observations of densities are as follows: $$ \begin{array}{lrrrrrr} \text { Location } & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline \text { Batter A } & .135 & .102 & .098 & .141 & .131 & .144 \\ \hline \text { Batter B } & .129 & .120 & .112 & .152 & .135 & .163 \end{array} $$ a. Do the data present sufficient evidence to indicate a difference between the average densities of cakes prepared using the two types of batter? b. Construct a \(95 \%\) confidence interval for the difference between the average densities for the two mixes.

Find the critical value(s) of \(t\) that specify the rejection region in these situations: a. A two-tailed test with \(\alpha=.01\) and \(12 d f\) b. A right-tailed test with \(\alpha=.05\) and \(16 d f\) c. A two-tailed test with \(\alpha=.05\) and \(25 d f\) d. A left-tailed test with \(\alpha=.01\) and \(7 d f\)

The calcium (Ca) content of a powdered mineral substance was analyzed 10 times with the following percent compositions recorded: $$ \begin{array}{lllll} .0271 & .0282 & .0279 & .0281 & .0268 \\ .0271 & .0281 & .0269 & .0275 & .0276 \end{array} $$ a. Find a \(99 \%\) confidence interval for the true calcium content of this substance. b. What does the phrase "99\% confident" mean? c. What assumptions must you make about the sampling procedure so that this confidence interval will be valid? What does this mean to the chemist who is performing the analysis?

Independent random samples from two normal populations produced the variances listed here: $$ \begin{array}{cc} \text { Sample Size } & \text { Sample Variance } \\ \hline 16 & 55.7 \\ 20 & 31.4 \end{array} $$ a. Do the data provide sufficient evidence to indicate that \(\sigma_{1}^{2}\) differs from \(\sigma_{2}^{2}\) ? Test using \(\alpha=.05\). b. Find the approximate \(p\) -value for the test and interpret its value.

The EPA limit on the allowable discharge of suspended solids into rivers and streams is 60 milligrams per liter ( \(\mathrm{mg} /\) l) per day. A study of water samples selected from the discharge at a phosphate mine shows that over a long period, the mean daily discharge of suspended solids is \(48 \mathrm{mg} /\), but day-to-day discharge readings are variable. State inspectors measured the discharge rates of suspended solids for \(n=20\) days and found \(s^{2}=39(\mathrm{mg} / \mathrm{I})^{2}\). Find a \(90 \%\) confidence interval for \(\sigma^{2}\). Interpret your results.
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