To find the missing values in the ANOVA table, we can use the following relationships:
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Degrees of Freedom (df):
- For Treatment: df = k - 1, where k is the number of treatments.
- For Error: df = N - k, where N is the total number of observations and k is the number of treatments.
- Total df = N - 1.
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Sum of Squares (SS):
- Total SS = Treatment SS + Error SS.
- SS values can be calculated as:
- MS (Mean Square) = SS / df.
- Therefore, SS = MS * df.
-
F-statistic:
- F-statistic = MS(Treatment) / MS(Error).
From the provided ANOVA table, we can see that:
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For Treatment:
- SS = A
- df = 2
- MS = 43,024.778
- Fstat = 25.1754
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For Error:
- SS = 10,254
- df = C
- MS = E
-
Total:
- SS = B
- df = D
Steps to find missing values:
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Calculate Treatment SS (A): \[ A = MS_{Treatment} \times df_{Treatment} \] \[ A = 43,024.778 \times 2 \] \[ A = 86,049.556 \]
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Compute Total SS (B): \[ B = A + Error , SS = A + 10,254 \] \[ B = 86,049.556 + 10,254 = 96,303.556 \]
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Calculate Error df (C): Since we don’t have N, let’s express C in terms of N and k. We know: \[ Total , df = (N - 1) \] Since Total df = Treatment df + Error df: \[ C = N - k \] Given that Treatment df = 2, hence \( k = 3 \). Therefore: \[ Total , df = (N - 1) = (N - 3 + 2); \] So N = C + 1.
The missing information remains about N; let’s estimate the Error df (C) directly depending on how many total treatments we might have.
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Using F-statistic: Calculate Error MS (E): \[ E = \frac{SS_{Error}}{df_{Error}} \] \[ \text{Given that } msTreatment = 43024.778, \text{ and Fstat = 25.1754} \] \[ \frac{43,024.778}{E} = 25.1754 \rightarrow E = \frac{43,024.778}{25.1754} \approx 1,706.166 \]
Thus, using \( C \): \[ 10,254 = E \times C, {1,706.166 \times C = 10,254 }. \]
- Solving for C: \[ C = \frac{10,254}{1,706.166} \approx 6 \]
So \( C \) = 6.
Final Values:
- Treatment SS (A) = 86,049.556
- Error df (C) = 6
- Total SS (B) = 96,303.556
Now you have the value of C, which is 6.
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