Problem
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 $\left(\mu_{1}\right): n=20, \bar{x}=0.59, s=1.04$ Reduction in Pain, Level After Sham Treatment $\left(\mu_{2}\right): n=20, \bar{x}=0.55, s=1.58$ a. Use a 0.05 significance level to test the claim that those treated with magnets have a greater mean reduction in pain than those given a sham treatment (similar to a placebo). What are the null and alternative hypotheses? A. \[ \begin{array}{l} H_{0}: \mu_{1} \neq \mu_{2} \\ H_{1}: \mu_{1}<\mu_{2} \end{array} \] c. \[ \begin{array}{l} H_{0}: \mu_{1}=\mu_{2} \\ H_{1}: \mu_{1} \neq \mu_{2} \end{array} \] B. \[ \begin{array}{l} H_{0}: \mu_{1}<\mu_{2} \\ H_{1}: \mu_{1} \geq \mu_{2} \end{array} \] D. \[ \begin{array}{l} H_{0}: \mu_{1}=\mu_{2} \\ H_{1}: \mu_{1}>\mu_{2} \end{array} \] The test statistic, $t$, is (Round to two decimal places as needed.)
Solution
Step 1 ：The null and alternative hypotheses are: \[H_{0}: \mu_{1}=\mu_{2}\] \[H_{1}: \mu_{1}>\mu_{2}\]
Step 2 ：The formula for the t statistic is: \[t = \frac{(\bar{x}_1 - \bar{x}_2)}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}}\]
Step 3 ：Substitute the given values into the formula: \[t = \frac{(0.59 - 0.55)}{\sqrt{\frac{1.04^2}{20} + \frac{1.58^2}{20}}}\]
Step 4 ：Calculate the t statistic to get: \[t = 0.09457031182808971\]
Step 5 ：Round the t statistic to two decimal places to get: \[t = 0.09\]
Step 6 ：The final answer is: \[\boxed{0.09}\]
