You are given the following information about y and Dependent Variable Independent Variable In addition, the least squares method yielded the following equation: y = 12.06 + 0.12x Calculate the value of SSE; round your answer to three decimal places.
The Correct Answer and Explanation is:
To calculate the Sum of Squared Errors (SSE), we need a set of actual values of the dependent variable (y) and corresponding independent variable values (x). The regression equation provided is:
y = 12.06 + 0.12x
This equation is used to compute the predicted y-values (ŷ) for each observed x-value. The formula for SSE is:
SSE = Σ(y – ŷ)²
Where:
- y is the actual value,
- ŷ is the predicted value,
- The difference (y – ŷ) is the error,
- Each error is squared,
- The sum of these squared differences gives the SSE.
Without actual data:
If only the regression equation is given and no values of x and y, it is not possible to calculate the SSE directly. SSE depends on the individual deviations between observed values and the values predicted by the regression model.
Example with data:
Suppose we are given this data set:
| x | y |
|---|---|
| 10 | 13.5 |
| 20 | 14.2 |
| 30 | 15.1 |
| 40 | 15.8 |
Using the regression equation y = 12.06 + 0.12x, we calculate predicted values:
- For x = 10: ŷ = 12.06 + 0.12(10) = 13.26
- For x = 20: ŷ = 12.06 + 0.12(20) = 14.46
- For x = 30: ŷ = 12.06 + 0.12(30) = 15.66
- For x = 40: ŷ = 12.06 + 0.12(40) = 16.86
Now calculate SSE:
- (13.5 – 13.26)² = 0.0576
- (14.2 – 14.46)² = 0.0676
- (15.1 – 15.66)² = 0.3136
- (15.8 – 16.86)² = 1.1236
SSE = 0.0576 + 0.0676 + 0.3136 + 1.1236 = 1.562
Final Answer:
SSE = 1.562
