The problem asks us to determine whether to reject or fail to reject the null hypothesis based on the given ANOVA table and to calculate the effect size. The ANOVA table provides the sums of squares (SS), degrees of freedom (df), mean squares (MS), and the F statistic.

1. Problem Description

The problem asks us to determine whether to reject or fail to reject the null hypothesis based on the given ANOVA table and to calculate the effect size. The ANOVA table provides the sums of squares (SS), degrees of freedom (df), mean squares (MS), and the F statistic.

2. Solution Steps

First, we determine whether to reject the null hypothesis. The F-statistic is given as $F = 5.83$. The degrees of freedom for the numerator (between groups) is $df_{between} = 2$, and the degrees of freedom for the denominator (within groups) is $df_{within} = 41$.

To make a decision, we need to compare the calculated F-statistic to a critical F-value at a chosen significance level (alpha). Since the alpha level is not specified, we will commonly use $\alpha = 0.05$. We look up the critical F-value for $df_{between} = 2$ and $df_{within} = 41$ at $\alpha = 0.05$. Using an F-distribution table or calculator, $F_{critical}(2, 41, 0.05) \approx 3.22$.

Since the calculated F-statistic ($5.83$) is greater than the critical F-value ($3.22$), we reject the null hypothesis.

Next, we calculate the effect size, specifically eta-squared ($\eta^2$), which represents the proportion of variance in the dependent variable that is explained by the independent variable. The formula for eta-squared is:

$\eta^2 = \frac{SS_{between}}{SS_{total}}$

From the ANOVA table, we have $SS_{between} = 60.72$ and $SS_{total} = 274.33$. Plugging these values into the formula, we get:

$\eta^2 = \frac{60.72}{274.33} \approx 0.221$

3. Final Answer

We reject the null hypothesis. The effect size, measured by eta-squared, is approximately 0.

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