A one-factor fixed-effects ANOVA is performed on data for 5 groups of unequal sizes, and H₀ is rejected at the .05 level of significance. Using the Scheffé procedure, test the contrast that
$\bar{Y}_{.1}-\frac{\bar{Y}_{.2}+\bar{Y}_{.3}}{2}$ at the .05 level of significance given the following information:
df_with = 30, = 5, n₁ = 6, = 7.5, n₂ = 6, = 8.5, n₃ = 6, and MS_with = 9.

Question

A one-factor fixed-effects ANOVA is performed on data for 5 groups of unequal sizes, and H₀ is rejected at the .05 level of significance. Using the Scheffé procedure, test the contrast that

\bar{Y}_{.1}-\frac{\bar{Y}_{.2}+\bar{Y}_{.3}}{2}

at the .05 level of significance given the following information:
df_with = 30, = 5, n₁ = 6, = 7.5, n₂ = 6, = 8.5, n₃ = 6, and MS_with = 9.

Quizplus · Accepted Answer

The Answer is For the contrast, $For the contrast, c1 = 1, c2 = -1/2, c3 = -1/2; n1 = n2 = n3 = 6; dfwith = 30; MSwith = 9; = 5, = 7.5, = 8.5; \alpha = .05; J = 5. Therefore, contrast \Psi = 5 -(7.5 + 8.5)/2 = -3; standard error of contrast: s \Psi = \sqrt{M S_{ ext {with }}\left(\frac{c_{1}^{2}}{n_{1}}+\frac{c_{2}^{2}}{n_{2}}+\frac{c_{3}^{2}}{n_{3}} ight)}=\sqrt{9\left(\frac{1}{6}+\frac{1}{24}+\frac{1}{24} ight)}=3 / 2 t = \Psi \Psi = -3/(3/2) = -2. critical values: Because |t| = 2 < 3.28, we fail to reject the null hypothesis and conclude that the contrast is not significant at the .05 level of significance.$
c₁= 1, c₂= -1/2, c₃= -1/2;
n₁= n₂ = n₃ = 6; df_with = 30; MS_with = 9; = 5, = 7.5, = 8.5;

\alpha

= .05; J = 5.

Therefore, contrast

\Psi

= 5 -(7.5 + 8.5)/2 = -3;
standard error of contrast: s_$\Psi$ =

\sqrt{M S_{ ext {with }}\left(\frac{c_{1}^{2}}{n_{1}}+\frac{c_{2}^{2}}{n_{2}}+\frac{c_{3}^{2}}{n_{3}} ight)}=\sqrt{9\left(\frac{1}{6}+\frac{1}{24}+\frac{1}{24} ight)}=3 / 2

t =

\Psi

_$\Psi$= -3/(3/2) = -2.

critical values:
$For the contrast, c1 = 1, c2 = -1/2, c3 = -1/2; n1 = n2 = n3 = 6; dfwith = 30; MSwith = 9; = 5, = 7.5, = 8.5; \alpha = .05; J = 5. Therefore, contrast \Psi = 5 -(7.5 + 8.5)/2 = -3; standard error of contrast: s \Psi = \sqrt{M S_{ ext {with }}\left(\frac{c_{1}^{2}}{n_{1}}+\frac{c_{2}^{2}}{n_{2}}+\frac{c_{3}^{2}}{n_{3}} ight)}=\sqrt{9\left(\frac{1}{6}+\frac{1}{24}+\frac{1}{24} ight)}=3 / 2 t = \Psi \Psi = -3/(3/2) = -2. critical values: Because |t| = 2 < 3.28, we fail to reject the null hypothesis and conclude that the contrast is not significant at the .05 level of significance.$

Because |t| = 2 < 3.28, we fail to reject the null hypothesis and conclude that the contrast is not significant at the .05 level of significance.

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A One-Factor Fixed-Effects ANOVA Is Performed on Data for 5 $\bar{Y}_{.1}-\frac{\bar{Y}_{.2}+\bar{Y}_{.3}}{2}$