Questions: Homework (Ch 04) Complete the following worksheet and then use it to calculate the coefficient of determination. The coefficient of determination (r^2) is The F-ratio is , which means that the assessor reject, at the 5% level of significance, the null hypothesis that there is relationship between the selling price and the area of the house. (Hint: The critical value of F0.05,1,13 is 4.67.) Which of the following is an approximate 95% prediction interval for the selling price of a house having an area (size) of 15 (hundred) square feet?

Solution Steps

The coefficient of determination, denoted as \( r^2 \), is calculated as follows:

\[ r^2 = 0.8907 \]

This indicates that approximately \( 89.07\% \) of the variability in the dependent variable \( y \) can be explained by the independent variable \( x \).

The F-ratio is calculated to test the significance of the regression model:

\[ F = 105.9086 \]

Since the critical value of \( F \) at the \( 5\% \) level of significance for \( (1, 13) \) degrees of freedom is \( 4.67 \), we compare:

\[ F > 4.67 \]

Thus, we reject the null hypothesis, indicating that there is a significant relationship between the selling price and the area of the house.

The means of \( x \) and \( y \) are calculated as follows:

\[ \bar{x} = 23.8467, \quad \bar{y} = 300.72 \]

The correlation coefficient \( r \) is:

\[ r = 0.9441 \]

The slope \( \beta \) is calculated using:

\[ \beta = \frac{\sum_{i=1}^{n} x_i y_i - n \bar{x} \bar{y}}{\sum_{i=1}^{n} x_i^2 - n \bar{x}^2} = \frac{2563.456}{652.6773} = 3.9276 \]

The intercept \( \alpha \) is calculated as:

\[ \alpha = \bar{y} - \beta \bar{x} = 300.72 - 3.9276 \times 23.8467 = 207.0598 \]

The equation of the line of best fit is:

\[ y = 207.0598 + 3.9276x \]

To predict the selling price for a house with an area of \( 15 \) (hundred) square feet, we substitute \( x = 15 \) into the regression equation:

\[ \text{Predicted } y = 207.0598 + 3.9276 \times 15 = 265.9738 \]

Final Answer