You wish to test the following claim $\left(H_{a}\right)$ at a significance level of $\alpha=0.005$. For the context ot 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 $=$
Answer
Final Answer: The critical value for this test is \(\boxed{2.576}\).
Step 1 ：Given that the significance level \(\alpha=0.005\) and the test is one-tailed (as we are testing \(\mu_{d}>0\)), we can use the standard normal distribution (Z-distribution) to find the critical value. The critical value is the z-score such that the area to its right under the standard normal curve is equal to \(\alpha\).
Step 2 ：Using the standard normal distribution table or a calculator, we find that the critical value is approximately 2.576.
Step 3 ：Final Answer: The critical value for this test is \(\boxed{2.576}\).
