You wish to test the following claim ($\displaystyle {H}_{{a}}$) at a significance level of $\displaystyle \alpha={0.001}$.

      $\displaystyle {H}_{{o}}:\mu_{{1}}=\mu_{{2}}$
      $\displaystyle {H}_{{a}}:\mu_{{1}}\ne\mu_{{2}}$

You believe both populations are normally distributed, but you do not know the standard deviations for either. However, you also have no reason to believe the variances of the two populations are not equal. You obtain a sample of size $\displaystyle {n}_{{1}}={23}$ with a mean of $\displaystyle {M}_{{1}}={68.6}$ and a standard deviation of $\displaystyle {S}{D}_{{1}}={9.8}$ from the first population. You obtain a sample of size $\displaystyle {n}_{{2}}={23}$ with a mean of $\displaystyle {M}_{{2}}={78.6}$ and a standard deviation of $\displaystyle {S}{D}_{{2}}={11.9}$ from the second population.

What is the critical value for this test? For this calculation, use the conservative under-estimate for the degrees of freedom as mentioned in the textbook. (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...