Question:
A regression equation π = β2.5 + 2π is estimated using the following data:
π 2 5 9 14 π 2 4 6 8
The coefficient of determination is ________ (rounded off to two decimal places).
A regression equation π = β2.5 + 2π is estimated using the following data:
The coefficient of determination is ________ (rounded off to two decimal places).
| π | 2 | 5 | 9 | 14 |
| π | 2 | 4 | 6 | 8 |
The coefficient of determination is ________ (rounded off to two decimal places).
Updated On: Feb 10, 2025
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Correct Answer: 0.98
Solution and Explanation
Calculating the Coefficient of Determination \( R^2 \)
Step 1: Formula for \( R^2 \)
The coefficient of determination is given by:
\[ R^2 = 1 - \frac{SS_{\text{residual}}}{SS_{\text{total}}} \]
Step 2: Calculate \( \bar{Y} \) (Mean of Y values)
\[ \bar{Y} = \frac{\sum Y_i}{n} = \frac{2 + 5 + 9 + 14}{4} \]
\[ = \frac{30}{4} = 7.5 \]
Step 3: Calculate \( SS_{\text{total}} \)
\[ SS_{\text{total}} = \sum (Y_i - \bar{Y})^2 \]
\[ SS_{\text{total}} = (2 - 7.5)^2 + (5 - 7.5)^2 + (9 - 7.5)^2 + (14 - 7.5)^2 \]
\[ = (-5.5)^2 + (-2.5)^2 + (1.5)^2 + (6.5)^2 \]
\[ = 30.25 + 6.25 + 2.25 + 42.25 = 81 \]
Step 4: Calculate Predicted Values \( \hat{Y}_i \)
Using the regression equation:
\[ \hat{Y} = -2.5 + 2X \]
We compute:
- \( \hat{Y}_1 = -2.5 + 2(2) = 1.5 \)
- \( \hat{Y}_2 = -2.5 + 2(4) = 5.5 \)
- \( \hat{Y}_3 = -2.5 + 2(6) = 9.5 \)
- \( \hat{Y}_4 = -2.5 + 2(8) = 13.5 \)
Step 5: Calculate \( SS_{\text{residual}} \)
\[ SS_{\text{residual}} = \sum (Y_i - \hat{Y}_i)^2 \]
\[ SS_{\text{residual}} = (2 - 1.5)^2 + (5 - 5.5)^2 + (9 - 9.5)^2 + (14 - 13.5)^2 \]
\[ = (0.5)^2 + (-0.5)^2 + (-0.5)^2 + (0.5)^2 \]
\[ = 0.25 + 0.25 + 0.25 + 0.25 = 1 \]
Step 6: Calculate \( R^2 \)
\[ R^2 = 1 - \frac{SS_{\text{residual}}}{SS_{\text{total}}} \]
\[ R^2 = 1 - \frac{1}{81} \]
\[ R^2 = 1 - 0.0123 = 0.9877 \approx 0.98 \]
Final Answer:
The coefficient of determination is \( R^2 = 0.98 \).
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