Step 1 :The null and alternative hypotheses for a statistical test comparing the means of two groups are typically a statement of no effect or no difference, and what we are testing for, respectively. In this case, we are testing whether the mean reduction in pain is greater for those treated with magnets than those given a sham treatment.
Step 2 :The null hypothesis is \(H_{0}: \mu_{1}=\mu_{2}\), which states that there is no difference in the mean reduction in pain between the two treatments.
Step 3 :The alternative hypothesis is \(H_{1}: \mu_{1}>\mu_{2}\), which states that the mean reduction in pain is greater for those treated with magnets.
Step 4 :The test statistic, t, can be calculated using the formula for the t statistic in a two-sample t test. This involves the sample means, sample sizes, and sample standard deviations of the two groups.
Step 5 :Given that the sample sizes (n1 and n2) are both 16, the sample means (\(x_{\bar{1}}\) and \(x_{\bar{2}}\)) are 0.55 and 0.37 respectively, and the sample standard deviations (s1 and s2) are 0.78 and 1.19 respectively, we can substitute these values into the formula.
Step 6 :The calculated t statistic is approximately 0.51 when rounded to two decimal places.
Step 7 :Final Answer: The null and alternative hypotheses are \(H_{0}: \mu_{1}=\mu_{2}\) and \(H_{1}: \mu_{1}>\mu_{2}\). The test statistic, t, is \(\boxed{0.51}\).