To fill in the missing values (A), (B), (C), (E), (F), and (G) in the ANOVA table, we first need to understand some key relationships in ANOVA calculations.
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Total Sum of Squares (SS Total): \[ SS_{\text{Total}} = SS_{\text{Treatment}} + SS_{\text{Error}} \] Here, \( SS_{\text{Treatment}} = 12,689 \) and we need to find \( SS_{\text{Error}} (A) \) and \( SS_{\text{Total}} (B) \).
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Degrees of Freedom (df): The degrees of freedom can be calculated as follows: \[ df_{\text{Treatment}} = k - 1 \quad \text{(where \( k \) is the number of treatments)} \] \[ df_{\text{Error}} = N - k \quad \text{(where \( N \) is the total number of observations)} \] The total degrees of freedom is: \[ df_{\text{Total}} = N - 1 \] Given that \( df_{\text{Total}} = 11 \), we also learn that: \[ N - 1 = 11 \implies N = 12 \]
Since we know that \( df_{\text{Treatment}} + df_{\text{Error}} = df_{\text{Total}} \): \[ df_{\text{Treatment}} + df_{\text{Error}} = 11 \] Let \( df_{\text{Treatment}} = (C) \) and \( df_{\text{Error}} = (E) \). Then: \[ (C) + (E) = 11 \]
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Mean Squares (MS): Mean Squares are calculated as follows: \[ MS_{\text{Treatment}} = \frac{SS_{\text{Treatment}}}{df_{\text{Treatment}}} = \frac{12,689}{C} \] \[ MS_{\text{Error}} = \frac{SS_{\text{Error}}}{df_{\text{Error}}} = \frac{A}{E} \]
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F-statistic: The F-statistic is calculated as follows: \[ F = \frac{MS_{\text{Treatment}}}{MS_{\text{Error}}} \]
Given \( F = 10.6642 \) and \( MS_{\text{Error}} = 595.3611 \): \[ MS_{\text{Treatment}} = F \times MS_{\text{Error}} = 10.6642 \times 595.3611 \] \[ MS_{\text{Treatment}} \approx 6348.688 \] 5. Finding Treatment Degrees of Freedom (C): Now we can find \( C \): \[ C = \frac{SS_{\text{Treatment}}}{MS_{\text{Treatment}}} = \frac{12,689}{6348.688} \approx 2 \]
Thus: \[ E = 11 - C = 11 - 2 = 9 \]
Summary Values:
- \( C = 2 \)
- \( E = 9 \)
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Finding SS Error (A): \[ MS_{\text{Error}} = \frac{A}{E} \implies 595.3611 = \frac{A}{9} \implies A = 595.3611 \times 9 \approx 5,358.250 \]
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Finding Total SS (B): \[ SS_{\text{Total}} = SS_{\text{Treatment}} + SS_{\text{Error}} = 12,689 + 5,358.250 \approx 18,047.250 \]
Results:
- (A) = 5,358.250 (approximately)
- (B) = 18,047.250 (approximately)
- (C) = 2
- (E) = 9
- (F) = 10.6642 (from the question)
- (G) = 10.6642 (same as F)
Now, you can fill in:
- (A): 5,358.250
- (B): 18,047.250
- (C): 2
- (E): 9
- (F): 10.6642
- (G): 10.6642