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 $\overline{d}=32.3$ 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

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 $\boxed{{\displaystyle 2.576}}$.

Step 2 ：The test statistic is calculated using the formula: $$t=\frac{\overline{d}-{\mu }_d}{s_d/\sqrt{n}}$$ where $\overline{d}$ 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, $\overline{d}=32.3$, ${\mu }_d=0$ (under the null hypothesis), $s_d=39.9$, and $n=12$. The test statistic for this sample is approximately $\boxed{{\displaystyle 2.804}}$.

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 $\boxed{{\displaystyle inthecriticalregion}}$.

Step 4 ：This test statistic leads to a decision to $\boxed{{\displaystyle rejectthenull}}$.