You wish to test the following claim $\left(H_{a}\right)$ at a significance level of $\alpha=0.005$. For the context of this problem, $\mu_{d}=\mu_{2}-\mu_{1}$ where the first data set represents a pre-test and the second data set represents a post-test. \[ \begin{array}{l} H_{o}: \mu_{d}=0 \\ H_{a}: \mu_{d}>0 \end{array} \] You believe the population of difference scores is normally distributed, but you do not know the standard deviation. You obtain pre-test and post-test samples for $n=12$ subjects. The average difference (post pre) is $\bar{d}=32.3$ with a standard deviation of the differences of $s_{d}=39.9$. What is the critical value for this test? (Report answer accurate to three decimal places.) critical value $=$ What is the test statistic for this sample? (Report answer accurate to three decimal places.) test statistic $=$ The test statistic is... in the critical region not in the critical region This test statistic leads to a decision to... reject the null accept the null fail to reject the null
Solution
Step 1 ：The critical value is the point beyond which we reject the null hypothesis. Since we are dealing with a one-tailed test (as indicated by \(H_{a}: \mu_{d}>0\)), we will use the z-score associated with our significance level \(\alpha=0.005\) to find the critical value. The critical value for this test is approximately \(\boxed{2.576}\).
Step 2 ：The test statistic is calculated using the formula: \[t = \frac{\bar{d} - \mu_{d}}{s_{d}/\sqrt{n}}\] where \(\bar{d}\) is the sample mean difference, \(\mu_{d}\) is the population mean difference under the null hypothesis, \(s_{d}\) is the standard deviation of the differences, and \(n\) is the sample size. In this case, \(\bar{d}=32.3\), \(\mu_{d}=0\) (under the null hypothesis), \(s_{d}=39.9\), and \(n=12\). The test statistic for this sample is approximately \(\boxed{2.804}\).
Step 3 ：Since the test statistic is greater than the critical value, the test statistic is in the critical region. This leads to a decision to reject the null hypothesis. The test statistic is \(\boxed{in the critical region}\).
Step 4 ：This test statistic leads to a decision to \(\boxed{reject the null}\).
