To calculate the total sum of squares (SSTotal) for the data provided, we first need to find the overall mean and then compute the sum of squares based on that mean. Here's how you can calculate it:
Step 1: Organize the data
Compact cars: 643, 655, 702
Midsize cars: 469, 427, 525
Full-size cars: 484, 456, 402
Step 2: Calculate the overall mean (Grand Mean)
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Find the total of all observations: \[ \text{Total} = 643 + 655 + 702 + 469 + 427 + 525 + 484 + 456 + 402 = 4563 \]
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Count the total number of observations: \[ n = 3 , (\text{compact}) + 3 , (\text{midsize}) + 3 , (\text{full size}) = 9 \]
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Calculate the overall mean: \[ \text{Grand Mean} (\bar{X}) = \frac{\text{Total}}{n} = \frac{4563}{9} = 507 \]
Step 3: Calculate SSTotal
SSTotal is calculated using the formula: \[ SSTotal = \sum (X_{ij} - \bar{X})^2 \] where \(X_{ij}\) represents each individual score and \(\bar{X}\) is the overall mean.
Now, we will calculate \((X_{ij} - \bar{X})^2\) for each score:
Compact cars: \[ (643 - 507)^2 = 18576 \] \[ (655 - 507)^2 = 21904 \] \[ (702 - 507)^2 = 36969 \]
Midsize cars: \[ (469 - 507)^2 = 144 \] \[ (427 - 507)^2 = 6400 \] \[ (525 - 507)^2 = 324 \]
Full-size cars: \[ (484 - 507)^2 = 529 \] \[ (456 - 507)^2 = 2601 \] \[ (402 - 507)^2 = 11025 \]
Step 4: Sum up these squared differences
Now, we can sum these squared differences:
\[ SSTotal = 18576 + 21904 + 36969 + 144 + 6400 + 324 + 529 + 2601 + 11025 \]
Calculating this gives:
\[ SSTotal = 18576 + 21904 + 36969 + 144 + 6400 + 324 + 529 + 2601 + 11025 = 93073 \]
Therefore, the total sum of squares (SSTotal) is 93,073.
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