Question
A chemical engineer is evaluating alternative temperature settings for one stage of processing a particular product. The three alternatives are low temperature, middle temperature, and high temperature. As blocking variables, three levels are each used for catalyst and solvent. The following processing times were obtained: / ‘n Chemiese ingeneur evalueer alternatiewe temperature verstellings vir een stadium van verwerking van ‘n sekere produk. Die drie alternatiewes is Laag-middel-en hoog, temperatuur. Die blok veranderlike verteenwoordig drie vlakke. Die volgende temperature word waargeneem.
Temperature Blocks Catalyst
(1) (2) (3)
Low 30 27 27 a1 2 352
Middle 26 26 23 a2 1 875
High 25 22 22 69 a3
81 a4 72 G
2 187 1 875 1 728
ANOVA TABLE / ANAVA TABEL
Source / Bron SS/ Sk df / vg MS/ GK F
Treatments / Behandeling
Blocks / Blokke 2
Residuals / Foute 1
Total / Totaal
CF = ?
Temperature Blocks Catalyst
(1) (2) (3)
Low 30 27 27 a1 2 352
Middle 26 26 23 a2 1 875
High 25 22 22 69 a3
81 a4 72 G
2 187 1 875 1 728
ANOVA TABLE / ANAVA TABEL
Source / Bron SS/ Sk df / vg MS/ GK F
Treatments / Behandeling
Blocks / Blokke 2
Residuals / Foute 1
Total / Totaal
CF = ?
Answers
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Answered by
GPT-4o mini
To calculate the Correction Factor (CF) in ANOVA, we use the formula:
\[
CF = \frac{(G^2)}{N}
\]
Where:
- \( G \) is the grand total of all observations.
- \( N \) is the total number of observations.
### Step 1: **Calculate the Grand Total (G)**
From the data provided, we can sum all the processing times:
- Low temperature: \( 30 + 27 + 27 = 84 \)
- Middle temperature: \( 26 + 26 + 23 = 75 \)
- High temperature: \( 25 + 22 + 22 = 69 \)
Now, summing all these values gives us the total:
\[
G = 84 + 75 + 69 = 228
\]
### Step 2: **Calculate the Total Number of Observations (N)**
In this scenario, you have a total of 3 temperatures and 3 blocks, making for a total of:
\[
N = 3 \ (temperatures) \times 3 \ (blocks) = 9
\]
### Step 3: **Calculate CF**
Now we can substitute \( G \) and \( N \) into the Correction Factor formula:
\[
CF = \frac{(G^2)}{N} = \frac{(228^2)}{9}
\]
Calculating \( G^2 \):
\[
228^2 = 51984
\]
Thus,
\[
CF = \frac{51984}{9} = 5776
\]
### Summary
The Correction Factor (CF) is:
\[
CF = 5776
\]
This value is necessary for completing the ANOVA table and calculating other components such as the Sums of Squares (SS). If you would like further assistance with the ANOVA calculations, feel free to ask!
\[
CF = \frac{(G^2)}{N}
\]
Where:
- \( G \) is the grand total of all observations.
- \( N \) is the total number of observations.
### Step 1: **Calculate the Grand Total (G)**
From the data provided, we can sum all the processing times:
- Low temperature: \( 30 + 27 + 27 = 84 \)
- Middle temperature: \( 26 + 26 + 23 = 75 \)
- High temperature: \( 25 + 22 + 22 = 69 \)
Now, summing all these values gives us the total:
\[
G = 84 + 75 + 69 = 228
\]
### Step 2: **Calculate the Total Number of Observations (N)**
In this scenario, you have a total of 3 temperatures and 3 blocks, making for a total of:
\[
N = 3 \ (temperatures) \times 3 \ (blocks) = 9
\]
### Step 3: **Calculate CF**
Now we can substitute \( G \) and \( N \) into the Correction Factor formula:
\[
CF = \frac{(G^2)}{N} = \frac{(228^2)}{9}
\]
Calculating \( G^2 \):
\[
228^2 = 51984
\]
Thus,
\[
CF = \frac{51984}{9} = 5776
\]
### Summary
The Correction Factor (CF) is:
\[
CF = 5776
\]
This value is necessary for completing the ANOVA table and calculating other components such as the Sums of Squares (SS). If you would like further assistance with the ANOVA calculations, feel free to ask!
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