This problem is an extension of problem 12.1 from Lomax & Hahs-Vaughn, 3rd ed.

A 1-way fixed-effects ANOVA is performed on data for 5 groups of equal sizes ($\displaystyle {n}={10}$ subjects in each group), and $\displaystyle {H}_{{0}}$ is rejected at the $\displaystyle \alpha={0.02}$ level of significance.  After thinking about the omnibus test and the nature of each level, the researcher decides to test these three orthogonal contrasts::

Contrast	$\displaystyle {c}_{{1}}$	$\displaystyle {c}_{{2}}$	$\displaystyle {c}_{{3}}$	$\displaystyle {c}_{{4}}$	$\displaystyle {c}_{{5}}$
$\displaystyle \psi_{{1}}$	$\displaystyle {2}$	$\displaystyle -{3}$	$\displaystyle {2}$	$\displaystyle -{3}$	$\displaystyle {2}$
$\displaystyle \psi_{{2}}$	$\displaystyle {1}$	$\displaystyle {0}$	$\displaystyle -{2}$	$\displaystyle {0}$	$\displaystyle {1}$
$\displaystyle \psi_{{3}}$	$\displaystyle {0}$	$\displaystyle {1}$	$\displaystyle {0}$	$\displaystyle -{1}$	$\displaystyle {0}$

Questions for reflection:  What is actually being tested with this contrast?  In particular, for each contrast, which combined group averages are being compared to what?  Finally, confirm that each contrast is valid and that each pair is orthogonal.

For this data, the error variance was estimated as $\displaystyle {M}{S}_{{\text{with}}}={55.9}$.  The sample means for each level of the factor were

Level	Mean
1	72.2
2	70.4
3	65.5
4	66.1
5	79.9

Using the Scheffé test, find the critical value for these contrasts (report to 3 decimal places):
$\displaystyle {t}_{{\text{c.v.}}}=$

For the first contrast:
$\displaystyle \psi_{{1}}=$       (report to 1 decimal place)
$\displaystyle {t}_{{1}}=$       (report to 3 decimal place)
Conclusion:

For the second contrast:
$\displaystyle \psi_{{2}}=$       (report to 1 decimal place)
$\displaystyle {t}_{{2}}=$       (report to 3 decimal place)
Conclusion:

For the third contrast:
$\displaystyle \psi_{{3}}=$       (report to 1 decimal place)
$\displaystyle {t}_{{3}}=$       (report to 3 decimal place)
Conclusion: