You wish to test the following claim ($\displaystyle {H}_{{a}}$) at a significance level of $\displaystyle \alpha={0.005}$.

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

You believe both populations are normally distributed, but you do not know the standard deviations for either. Use non-pooled test. You obtain a sample of size $\displaystyle {n}_{{1}}={24}$ with a mean of $\displaystyle {M}_{{1}}={86.8}$ and a standard deviation of $\displaystyle {S}{D}_{{1}}={16.2}$ from the first population. You obtain a sample of size $\displaystyle {n}_{{2}}={26}$ with a mean of $\displaystyle {M}_{{2}}={78.6}$ and a standard deviation of $\displaystyle {S}{D}_{{2}}={14.2}$ 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...