You wish to test the following claim ($\displaystyle {H}_{{a}}$) at a significance level of $\displaystyle \alpha={0.001}$. For the context of this problem, $\displaystyle \mu_{{d}}=\mu_{{2}}-\mu_{{1}}$ where the first data set represents a pre-test and the second data set represents a post-test.

      $\displaystyle {H}_{{o}}:\mu_{{d}}={0}$
      $\displaystyle {H}_{{a}}:\mu_{{d}}\ne{0}$

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 $\displaystyle {n}={150}$ subjects. The average difference (post - pre) is $\displaystyle \overline{{d}}=-{5}$ with a standard deviation of the differences of $\displaystyle {s}_{{d}}={27.9}$.

What is the critical value for this test? (Report answer accurate to three decimal places.)
critical value = $\displaystyle \pm$

What is the test statistic for this sample? (Report answer accurate to three decimal places.)
test statistic =

The test statistic is...

This test statistic leads to a decision to...

As such, the final conclusion is that...