To find the value of \( E \) in the ANOVA table, we need to complete the missing information based on the relationships between the various components of the ANOVA.
The relevant formulas are:
- \( MS = \frac{SS}{df} \) (Mean Square)
- \( F = \frac{MS_{treatment}}{MS_{error}} \) (F-statistic)
Given values:
- \( MS_{treatment} = 43,024.778 \)
- \( df_{treatment} = 2 \)
- \( SS_{error} = 10,254 \)
Step-by-step calculation:
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Calculate \( MS_{error} \): Since we have the SS for the error and need the degrees of freedom (df): \[ MS_{error} = \frac{SS_{error}}{df_{error}} \] We need to find \( df_{error} \) first to get \( MS_{error} \).
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Calculate \( df_{error} \): The total degrees of freedom is given by \( df_{total} = df_{treatment} + df_{error} \). Let’s denote \( df_{error} \) as \( C \). Thus, \[ df_{total} = 2 + C \]
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Calculate \( F \): The F-statistic is calculated as: \[ F = \frac{MS_{treatment}}{MS_{error}} \] Rearranging this gives: \[ MS_{error} = \frac{MS_{treatment}}{F} \] Plugging in the values we have: \[ F = 25.1754 \] \[ MS_{error} = \frac{43,024.778}{25.1754} = 1,706.847 \]
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Substitute \( MS_{error} \) to find \( E \): Now we turn to the definition of mean square to find \( E \): \[ MS_{error} = \frac{SS_{error}}{df_{error}} \rightarrow 1,706.847 = \frac{10,254}{C} \] Solving for \( C \): \[ C \cdot 1,706.847 = 10,254 \] \[ C = \frac{10,254}{1,706.847} \approx 6 \]
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Calculate \( E \): Now that we have \( C \), we can find \( E \): \[ E = MS_{error} = 1,706.847 \]
Thus, the value of \( E \) is approximately:
\[ \boxed{1,706.847} \]
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