To perform the ANOVA analysis for the given data, we first need to organize our information and calculate the necessary components such as the mean for each treatment, the sums of squares (SS), and the degrees of freedom (df).
Step 1: Understand the Data and Definitions
We have three temperatures (low, middle, high) and the results for each of these temperatures across three levels of a blocking variable (catalyst levels). The processing times are:
- Low (30, 27, 27)
- Middle (26, 26, 23)
- High (25, 22, 22)
Step 2: Calculate Treatments for Each Temperature Level
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Mean Calculation for each Temperature:
- Low temperature mean: \( \text{Mean(low)} = \frac{30 + 27 + 27}{3} = \frac{84}{3} = 28 \)
- Middle temperature mean: \( \text{Mean(middle)} = \frac{26 + 26 + 23}{3} = \frac{75}{3} = 25 \)
- High temperature mean: \( \text{Mean(high)} = \frac{25 + 22 + 22}{3} = \frac{69}{3} = 23 \)
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Overall Mean: \[ \text{Overall Mean} = \frac{84 + 75 + 69}{9} = \frac{228}{9} = 25.3333 \]
Step 3: Calculate Sums of Squares
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Treatment Sum of Squares (SST): \[ SST = n \sum ( \text{mean}_i - \text{overall mean})^2 \] where \( n = \text{number of observations per treatment} \) (which is 3 in this case): \[ SST = 3 \left( (28 - 25.3333)^2 + (25 - 25.3333)^2 + (23 - 25.3333)^2 \right) \] \[ = 3 \left( (2.6667)^2 + (-0.3333)^2 + (-2.3333)^2 \right) \] \[ = 3 \left( 7.1111 + 0.1111 + 5.4444 \right) = 3 \times 12.6666 = 38.000 \]
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Block Sum of Squares (SSB): Here, if the blocking variable means something specific, we would compute it based on varying catalyst levels. If you are given specific data for blocks, we would proceed similarly to treatments.
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Residuals (SSR): \[ SSR = SST - SSB \]
Step 4: Fill in the ANOVA Table
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Degrees of Freedom (df) for Treatments: \( k - 1 \) \[ df_T = 3 - 1 = 2 \]
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Degrees of Freedom for Blocks (assuming 3 catalyst levels): \[ df_B = 3 - 1 = 2 \]
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Total df: \( N - 1 = 9 - 1 = 8 \)
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Residual degrees of freedom: \[ df_R = df_T + df_B - df = 2 + 2 - 8 = 2 \]
Using these components, we can set up the ANOVA table:
| Source | SS | df | MS | F | |--------------|----------|----|---------------|------| | Treatments | 38.00 | 2 | 19.00 | F | | Blocks | SSB | 2 | MSB | F_B | | Residuals | SSR | 4 | MSR | | | Total | Total SS| 8 | | |
Final Steps
You need to fill in the calculated sums of squares, compute mean squares, and ultimately compute the F-statistic for comparisons. This requires an understanding of the residuals if blocks are specified and variance calculations as per specific requirements.
Let me know if you have any question about the process or if you need any specific calculations or interpretations!