Researchers conducted a study to determine whether magnets are effective in treating back pain. Pain was measured using the visual analog scale, and the results shown below are among the results obtained in the study. Higher scores correspond to greater pain levels. Assume that the two samples are independent simple random samples selected from normally distributed populations, and do not assume that the population standard deviations are equal. Complete parts (a) to (c) below.
Reduction in Pain Level After Magnet Treatment $\left(\mu_{1}\right): n=19, \bar{x}=0.44, s=0.94$
Reduction in Pain Level After Sham Treatment $\left(\mu_{2}\right): n=19, \bar{x}=0.36, s=1.31$
State the conclusion for the test.
the null hypothesis. There sufficient evidence to support the claim that those treated with magnets have a greater mean reduction in pain than those given a sham treatment.
b. Construct a confidence interval appropriate for the hypothesis test in part (a).
\[
\square< \mu_{1}-\mu_{2}< \square
\]
Answer
a. The null hypothesis for this test would be that there is no difference in the mean reduction in pain between those treated with magnets and those given a sham treatment. This is represented as H0: μ1 = μ2. The alternative hypothesis would be that there is a difference in the mean reduction in pain between the two treatments, represented as Ha: μ1 ≠ μ2. Given the data, we can't conclude that there is sufficient evidence to support the claim that those treated with magnets have a greater mean reduction in pain than those given a sham treatment. The mean reduction in pain for the magnet treatment group (0.44) is only slightly higher than the mean reduction in pain for the sham treatment group (0.36), and the standard deviations for the two groups overlap, indicating a high degree of variability in the data. b. To construct a confidence interval for the difference in means, we would use the formula for a confidence interval for two independent samples: CI = (μ1 - μ2) ± t * √[(s1^2/n1) + (s2^2/n2)] where t is the t-score for the desired level of confidence, s1 and s2 are the standard deviations of the two samples, and n1 and n2 are the sample sizes. Assuming a 95% confidence level, the t-score for a two-tailed test with 36 degrees of freedom (n1 + n2 - 2) is approximately 2.028. Substituting the given values into the formula, we get: CI = (0.44 - 0.36) ± 2.028 * √[(0.94^2/19) + (1.31^2/19)] CI = 0.08 ± 2.028 * √[(0.0463) + (0.0901)] CI = 0.08 ± 2.028 * √0.1364 CI = 0.08 ± 2.028 * 0.3693 CI = 0.08 ± 0.749 So, the 95% confidence interval for the difference in means is (-0.669, 0.829). Since this interval includes zero, we cannot reject the null hypothesis that there is no difference in the mean reduction in pain between the two treatments.
Step 1 ：a. The null hypothesis for this test would be that there is no difference in the mean reduction in pain between those treated with magnets and those given a sham treatment. This is represented as H0: μ1 = μ2. The alternative hypothesis would be that there is a difference in the mean reduction in pain between the two treatments, represented as Ha: μ1 ≠ μ2. Given the data, we can't conclude that there is sufficient evidence to support the claim that those treated with magnets have a greater mean reduction in pain than those given a sham treatment. The mean reduction in pain for the magnet treatment group (0.44) is only slightly higher than the mean reduction in pain for the sham treatment group (0.36), and the standard deviations for the two groups overlap, indicating a high degree of variability in the data. b. To construct a confidence interval for the difference in means, we would use the formula for a confidence interval for two independent samples: CI = (μ1 - μ2) ± t * √[(s1^2/n1) + (s2^2/n2)] where t is the t-score for the desired level of confidence, s1 and s2 are the standard deviations of the two samples, and n1 and n2 are the sample sizes. Assuming a 95% confidence level, the t-score for a two-tailed test with 36 degrees of freedom (n1 + n2 - 2) is approximately 2.028. Substituting the given values into the formula, we get: CI = (0.44 - 0.36) ± 2.028 * √[(0.94^2/19) + (1.31^2/19)] CI = 0.08 ± 2.028 * √[(0.0463) + (0.0901)] CI = 0.08 ± 2.028 * √0.1364 CI = 0.08 ± 2.028 * 0.3693 CI = 0.08 ± 0.749 So, the 95% confidence interval for the difference in means is (-0.669, 0.829). Since this interval includes zero, we cannot reject the null hypothesis that there is no difference in the mean reduction in pain between the two treatments.
