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Chapter 9: Problem 11

Researchers conducted a study to determine whether magnets are effective in treating back pain, with results given below (based on data from "Bipolar Permanent Magnets for the Treatment of Chronic Lower Back Pain: A Pilot Study." by Collacott, Zimmerman, White, and Rindone, Journal of the American Medical Association, Vol. 283, No. 10. The values represent measurements of pain using the visual analog scale. Use a 0.05 significance level to test the claim that those given a sham treatment (similar to a placebo) have pain reductions that vary more than the pain reductions for those treated with magnets. Reduction in Pain Level After Sham Treatment: \(\quad n=20, \bar{x}=0.44, s=1.4\) Reduction in Pain Level After Magnet Treatment: \(\quad n=20, \bar{x}=0.49, s=0.96\)

Short Answer

Expert verified
Fail to reject the null hypothesis; not enough evidence to support the claim.

Step by step solution

01

State the Hypotheses

State the null and alternative hypotheses. \( H_0: \sigma^2_{sham} = \sigma^2_{magnet} \) \( H_1: \sigma^2_{sham} > \sigma^2_{magnet} \)
02

Significance Level

The significance level is given as \( \alpha = 0.05 \).
03

Calculate the Test Statistic

Use the F-test for variance. The test statistic is calculated as \[ F = \frac{s^2_{sham}}{s^2_{magnet}} \] \[ F = \frac{1.4^2}{0.96^2} = \frac{1.96}{0.9216} \approx 2.13 \]
04

Degrees of Freedom

Determine the degrees of freedom for both variances. \[ df_1 = n_1 - 1 = 20 - 1 = 19 \] \[ df_2 = n_2 - 1 = 20 - 1 = 19 \]
05

Determine the Critical Value

Find the critical value from the F-distribution table at \( \alpha = 0.05 \) for \( df_1 = 19 \) and \( df_2 = 19 \). The critical value \( F_{crit} \) is approximately 2.54.
06

Compare Test Statistic to Critical Value

Compare the calculated test statistic to the critical value. Since \( F = 2.13 \) is less than \( F_{crit} = 2.54 \), we fail to reject the null hypothesis.
07

Make a Decision

Since we fail to reject the null hypothesis, at the 0.05 significance level, there isn't enough evidence to support the claim that the pain reductions with sham treatment vary more than those with magnet treatment.

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Key Concepts

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

F-test
The F-test is a statistical method used to compare two variances to see if they are significantly different from each other. In this study, we use the F-test to compare the variance in pain reduction between two groups: one treated with a sham treatment and the other with magnets. A key component of the F-test is the F-statistic, calculated as the ratio of the variances of the two groups. Specifically, you calculate the F-statistic using the formula 聽\[ F = \frac{s^2_{sham}}{s^2_{magnet}} \].

This test helps determine whether any observed differences in variance are due to random chance or a real effect from the treatments.
Null Hypothesis
In hypothesis testing, the null hypothesis is a statement that there is no effect or no difference. It is the hypothesis we initially assume to be true. For our example, the null hypothesis (聽\( H_0 \)) is that the variance in pain reduction for the sham-treated group is equal to the variance in pain reduction for the magnet-treated group.

Mathematically, it is stated as 聽\( \sigma^2_{sham} = \sigma^2_{magnet} \). We test this hypothesis by comparing the calculated F-statistic to a critical value from the F-distribution. If our test statistic falls within the acceptance region, we fail to reject the null hypothesis.
Variance Comparison
In this study, we are interested in comparing the variances of pain reductions between two groups to see if there is more variability in one group compared to the other. Variance is a measure of how much the values in a data set differ from the mean. Higher variance means there is greater variability among the data points.

For the sham-treated group, the calculated variance is \(s^2_{sham} = 1.4^2 = 1.96 \) and for the magnet-treated group, it is \( s^2_{magnet} = 0.96^2 = 0.9216 \).

We use the F-test to compare these variances, calculating the F-statistic as 聽\[ F = \frac{1.96}{0.9216} \approx 2.13 \]. The next steps involve comparing this value to the critical value from the F-distribution.
Significance Level
The significance level ( \( \alpha \) ) is a threshold set by the researcher to determine when to reject the null hypothesis. In this study, the significance level is \( \alpha = 0.05 \), which means there's a 5% risk of concluding that a difference exists when there is no actual difference.

To decide whether to reject the null hypothesis, we compare the calculated F-statistic to the critical value from the F-distribution table at the \( \alpha = 0.05 \) significance level and with the corresponding degrees of freedom ( \( df_1 = 19 \) and \( df_2 = 19 \)).

In this case, the critical value is approximately 2.54. Since our calculated F-statistic (2.13) is less than the critical value, we fail to reject the null hypothesis. This means there isn't enough evidence to support the claim that the sham treatment produces more variable pain reduction than the magnet treatment.

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Most popular questions from this chapter

Test the given claim. Identify the null hypothesis, alternative hypothesis, test statistic, P-value or critical value(s), then state the conclusion about the null hypothesis, as well as the final conclusion that addresses the original claim. A study investigated rates of fatalities among patients with serious traumatic injuries. Among 61,909 patients transported by helicopter, 7813 died. Among 161,566 patients transported by ground services, 17,775 died (based on data from "Association Between Helicopter vs Ground Emergency Medical Services and Survival for Adults With Major Trauma," by Galvagno et al., Journal of the American Medical Association, Vol. \(307,\) No. 15 ). Use a 0.01 significance level to test the claim that the rate of fatalities is higher for patients transported by helicopter. a. Test the claim using a hypothesis test. b. Test the claim by constructing an appropriate confidence interval. c. Considering the test results and the actual sample rates, is one mode of transportation better than the other? Are there other important factors to consider?

Use the listed paired sample data, and assume that the samples are simple random samples and that the differences have a distribution that is approximately normal. Listed below are brain volumes (cm \(^{3}\) ) of twins from Data Set 8 "IQ and Brain Size" in Appendix B. Construct a \(99 \%\) confidence interval estimate of the mean of the differences between brain volumes for the first-born and the second-born twins. What does the confidence interval suggest? $$\begin{array}{l|r|r|r|r|r|r|r|r|r|r} \hline \text { First Born } & 1005 & 1035 & 1281 & 1051 & 1034 & 1079 & 1104 & 1439 & 1029 & 1160 \\ \hline \text { Second Born } & 963 & 1027 & 1272 & 1079 & 1070 & 1173 & 1067 & 1347 & 1100 & 1204 \\ \hline \end{array}$$

Test the given claim. Researchers from the University of British Columbia conducted trials to investigate the effects of color on the accuracy of recall. Subjects were given tasks consisting of words displayed on a computer screen with background colors of red and blue. The subjects studied 36 words for 2 minutes, and then they were asked to recall as many of the words as they could after waiting 20 minutes. Results from scores on the word recall test are given below. Use a 0.05 significance level to test the claim that variation of scores is the same with the red background and blue background. $$\begin{array}{l|l} \hline \text { Red Background: } & n=35, \bar{x}=15.89, s=5.90 \\ \hline \text { Blue Background: } & n=36, \bar{x}=12.31, s=5.48 \\ \hline \end{array}$$

Test the given claim. Identify the null hypothesis, alternative hypothesis, test statistic, P-value or critical value(s), then state the conclusion about the null hypothesis, as well as the final conclusion that addresses the original claim. A study investigated survival rates for in-hospital patients who suffered cardiac arrest. Among 58,593 patients who had cardiac arrest during the day, 11,604 survived and were discharged. Among 28,155 patients who suffered cardiac arrest at night, 4139 survived and were discharged (based on data from "Survival from In-Hospital Cardiac Arrest During Nights and Weekends," by Peberdy et al., Joumal of the American Medical Association, Vol. 299, No. 7). We want to use a 0.01 significance level to test the claim that the survival rates are the same for day and night. a. Test the claim using a hypothesis test. b. Test the claim by constructing an appropriate confidence interval. c. Based on the results, does it appear that for in-hospital patients who suffer cardiac arrest, the survival rate is the same for day and night?

In the article "On Judging the Significance of Differences by Examining the Overlap Between Confidence Intervals," by Schenker and Gentleman (American Statistician, Vol. 55, No. 3), the authors consider sample data in this statement: "Independent simple random samples, each of size \(200\) , have been drawn, and 112 people in the first sample have the attribute, whereas 88 people in the second sample have the attribute." a. Use the methods of this section to construct a \(95 \%\) confidence interval estimate of the difference \(p_{1}-p_{2} .\) What does the result suggest about the equality of \(p_{1}\) and \(p_{2} ?\) b. Use the methods of Section \(7-1\) to construct individual \(95 \%\) confidence interval estimates for each of the two population proportions. After comparing the overlap between the two confidence intervals, what do you conclude about the equality of \(p_{1}\) and \(p_{2} ?\) c. Use a 0.05 significance level to test the claim that the two population proportions are equal. What do you conclude? d. Based on the preceding results, what should you conclude about the equality of \(p_{1}\) and \(p_{2} ?\) Which of the three preceding methods is least effective in testing for the equality of \(p_{1}\) and \(p_{2} ?\)
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