Question

A regression equation 𝑌 = −2.5 + 2𝑋 is estimated using the following data:

𝑌	2	5	9	14
𝑋	2	4	6	8

The coefficient of determination is ________ (rounded off to two decimal places).

Answer 1

Correct Answer: 0.98

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 \( R^2 = 0.98 \).