Questions: a) Complete the ANOVA table below. Source Sum of Squares Degrees of Freedom Mean Sum of Squares F --- --- --- --- --- Between 56 2 28.0 9.333 Within 27 9 3.0 Total 83 11 (Type integers or decimals rounded to three decimal places as needed.) b) There are 3 population means being tested. c) What are the hypotheses for this test? A. H0: None of the means are equal. H1: All the means are equal. C. H0: All the means are equal. H1: Not all the means are equal. Determine the critical F-score, Fα, for this test. Fα= (Round to three decimal places as needed.)

a) Complete the ANOVA table below. Source Sum of Squares Degrees of Freedom Mean Sum of Squares F --- --- --- --- --- Between 56 2 28.0 9.333 Within 27 9 3.0 Total 83 11 (Type integers or decimals rounded to three decimal places as needed.) b) There are 3 population means being tested. c) What are the hypotheses for this test? A. H0: None of the means are equal. H1: All the means are equal. C. H0: All the means are equal. H1: Not all the means are equal. Determine the critical F-score, Fα, for this test. Fα= (Round to three decimal places as needed.)
Transcript text: a) Complete the ANOVA table below. \begin{tabular}{lcccc} \hline Source & \begin{tabular}{c} Sum of \\ Squares \end{tabular} & \begin{tabular}{c} Degrees of \\ Freedom \end{tabular} & \begin{tabular}{c} Mean Sum of \\ Squares \end{tabular} & F \\ \hline Between & 56 & 2 & 28.0 & 9.333 \\ Within & 27 & 9 & 3.0 & \\ Total & 83 & 11 & & \\ \hline \end{tabular} (Type integers or decimals rounded to three decimal places as needed.) b) There are 3 population means being tested. c) What are the hypotheses for this test? A. $\mathrm{H}_{0}$ : None of the means are equal. $\mathrm{H}_{1}$ : All the means are equal. C. $\mathrm{H}_{0}$ : All the means are equal. $\mathrm{H}_{1}$ : Not all the means are equal. Determine the critical F -score, $\mathrm{F}_{\alpha}$, for this test. $\mathrm{F}_{\alpha}=$ $\square$ (Round to three decimal places as needed.)
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Solution

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Solution Steps

Step 1: Calculate Degrees of Freedom Total

The total degrees of freedom (\(df_{total}\)) is calculated as the sum of the degrees of freedom between groups (\(df_{between}\)) and within groups (\(df_{within}\)): \[ df_{total} = df_{between} + df_{within} = 2 + 9 = 11 \]

Step 2: Calculate Sum of Squares Total

The total sum of squares (\(SS_{total}\)) is the sum of the sum of squares between groups (\(SS_{between}\)) and within groups (\(SS_{within}\)): \[ SS_{total} = SS_{between} + SS_{within} = 56 + 27 = 83 \]

Step 3: State the Hypotheses

The hypotheses for the ANOVA test are defined as follows:

  • Null Hypothesis (\(H_0\)): All the means are equal.
  • Alternative Hypothesis (\(H_1\)): Not all the means are equal.

Final Answer

The calculated degrees of freedom total is \(11\), the calculated sum of squares total is \(83\), and the hypotheses are:

  • \(H_0: \mu_1 = \mu_2 = \mu_3\)
  • \(H_1: \text{At least one } \mu_i \text{ is different}\)

Thus, the final answer is: \[ \boxed{H_0: \text{All the means are equal; } H_1: \text{Not all the means are equal.}} \]

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