To fill in the values for the ANOVA table, we need to first understand some key relationships in ANOVA. Let's go through it step by step:
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Degrees of Freedom (df):
- For Treatments (df_treatments): Typically, df is one less than the number of treatment groups. Therefore, if we let \( d \) be the number of treatment groups, then: \[ \text{df treating} = d - 1 \]
- For Blocks: Already given as 5, which means there are 6 blocks (block df = 5).
- For Errors: df for errors is the total df minus the sum of df for treatments and blocks.
- Total df (df_total) is the total number of observations minus 1.
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Mean Squares (MS):
- Mean Square for Treatments (MS_treatments) is given as: \[ \text{MS Treatments} = \frac{SS Treatments}{df Treatments} \] \[ f = \frac{SS treatments}{d} = 35.6675 \]
- For Blocks: \[ \text{MS Blocks} = \frac{SS Blocks}{df Blocks} = \frac{68.14}{5} = 13.628 \]
- For Errors: \[ \text{MS Errors} = \frac{SS Errors}{df Errors} = 1.9165 \]
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Sum of Squares (SS):
- Total SS is the sum of Treatments, Blocks, and Errors: \[ \text{Total SS} = SS Treatments + SS Blocks + SS Errors \]
- From given data, we can denote: \[ c = 142.6 + 68.14 + b \]
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Calculating SS Errors: You have \( df_errors = 20 \). Since MS for errors is given as: \[ \text{MS Errors} = \frac{SS Errors}{df Errors} = 1.9165 \] Then: \[ b = MS Errors \times df Errors = 1.9165 \times 20 = 38.33 \]
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Finding Values:
- Value of (b): Now that we have \( b \): \[ b = 38.33 \]
- Finding Total SS (c): \[ c = 142.6 + 68.14 + 38.33 = 249.07 \]
- Finding Total df (e): Total df = Total number of samples (N) - 1. Assuming that N is the total number of observations across treatments and blocks.
Given that:
- If Treatments have \( d \) groups and Blocks have 6, then if we have:
For the formula: \[ N = df_treatments + df_blocks + df_errors + 1 = (d - 1) + 5 + 20 + 1 \] This would mean: \[ N = d + 25 \] Thus, the total df (e) for the ANOVA table would be: \[ e = N - 1 = (d + 25) - 1 = d + 24 \]
Assuming there are 6 treatment groups:
If \( d = 6 \): \[ e = 6 + 24 = 30 \]
Using these relations, we can fill out the missing values in the ANOVA table.
In conclusion:
- The value of (e) as calculated is: e = 30
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