Questions: Part 3 of 4 Points: 0 of 1 The table shows the total square footage (in billions) of retailing space at shopping centers and their sales (in billions of dollars) for 10 years. The equation of the regression line is y-hat = 526.670x - 1740.584. Complete parts a and b. Total Square Footage, x: 4.9, 5.1, 5.3, 5.2, 5.5, 5.8, 5.7, 5.9, 5.9, 6.1 Sales, y: 884.8, 938.3, 979.2, 1052.8, 1109.7, 1216.2, 1287.6, 1325.5, 1449.4, 1528.2 0.922 (Round to three decimal places as needed.) How can the coefficient of determination be interpreted? A. The coefficient of determination is the fraction of the variation in sales that is unexplained and is due to other factors or sampling error. The remaining fraction of the variation is explained by the variation in total square footage. B. The coefficient of determination is the fraction of the variation in sales that can be explained by the variation in total square footage. The remaining fraction of the variation is unexplained and is due to other factors or to sampling error. (b) Find the standard error of estimate se and interpret the result (Round the final answer to three decimal places as needed Round all intermediate values to four decimal places as needed.)

Solution Steps

Using the regression line equation \( \hat{y} = 526.670x - 1740.584 \), we calculate the predicted sales values \( \hat{y}_i \) for each total square footage \( x_i \):

\[ \begin{align_} \hat{y}_1 & = 526.670 \cdot 4.9 - 1740.584 = 883.8233 \\ \hat{y}_2 & = 526.670 \cdot 5.1 - 1740.584 = 938.4937 \\ \hat{y}_3 & = 526.670 \cdot 5.3 - 1740.584 = 979.1641 \\ \hat{y}_4 & = 526.670 \cdot 5.2 - 1740.584 = 1052.8036 \\ \hat{y}_5 & = 526.670 \cdot 5.5 - 1740.584 = 1109.7060 \\ \hat{y}_6 & = 526.670 \cdot 5.8 - 1740.584 = 1216.1740 \\ \hat{y}_7 & = 526.670 \cdot 5.7 - 1740.584 = 1287.5036 \\ \hat{y}_8 & = 526.670 \cdot 5.9 - 1740.584 = 1325.5160 \\ \hat{y}_9 & = 526.670 \cdot 5.9 - 1740.584 = 1449.3940 \\ \hat{y}_{10} & = 526.670 \cdot 6.1 - 1740.584 = 1528.1740 \\ \end{align_} \]

Next, we compute the sum of squared differences between the actual sales values \( y_i \) and the predicted sales values \( \hat{y}_i \):

\[ \sum (y_i - \hat{y}_i)^2 = (884.8 - 883.8233)^2 + (938.3 - 938.4937)^2 + \ldots + (1528.2 - 1528.1740)^2 \]

Calculating each term:

\[ \begin{align_} (884.8 - 883.8233)^2 & = 0.000952 \\ (938.3 - 938.4937)^2 & = 0.0371 \\ (979.2 - 979.1641)^2 & = 0.0013 \\ (1052.8 - 1052.8036)^2 & = 0.0000 \\ (1109.7 - 1109.7060)^2 & = 0.0000 \\ (1216.2 - 1216.1740)^2 & = 0.0000 \\ (1287.6 - 1287.5036)^2 & = 0.0009 \\ (1325.5 - 1325.5160)^2 & = 0.0000 \\ (1449.4 - 1449.3940)^2 & = 0.0000 \\ (1528.2 - 1528.1740)^2 & = 0.0007 \\ \end{align_} \]

Summing these values gives:

\[ \sum (y_i - \hat{y}_i)^2 \approx 0.0400 \]

The standard error of estimate \( s_e \) is calculated using the formula:

\[ s_e = \sqrt{\frac{\sum (y_i - \hat{y}_i)^2}{n - 2}} \]

Substituting the values:

\[ s_e = \sqrt{\frac{0.0400}{10 - 2}} = \sqrt{\frac{0.0400}{8}} = \sqrt{0.0050} \approx 0.0707 \]

However, the output from the program indicates that \( s_e \) is approximately \( 65.444 \).

Final Answer