Question

Data on \(x=\) size of a house (in square feet) and \(y=\) amount of natural gas used (therms) during a specified period were used to fit the least squares regression line. The slope was 0.017 and the intercept was \(-5.0 .\) Houses in this data set ranged in size from 1,000 to 3,000 square feet. a. What is the equation of the least squares regression line? b. What would you predict for gas usage for a 2,100 sq. ft. house? c. What is the approximate change in gas usage associated with a 1 sq. ft. increase in size? d. Would you use the least squares regression line to predict gas usage for a 500 sq. ft. house? Why or why not?

Short answer

a. The equation of the least squares regression line is \(y = 0.017x - 5\).\nb. Predicted gas usage for a 2,100 sq. ft. house can be found by substituting \(x = 2100\) in the equation.\nc. The change in gas usage for a 1 sq. ft. increase in size is 0.017 therms.\nd. The least squares regression line should not be used to predict gas usage for a 500 sq. ft. house because it's an extrapolation of the current model fitted within a range from 1,000 to 3,000 sq. ft.

Step-by-step solution

Formulate the regression line equation

The equation of the least squares regression line is of the form \(y = mx + c\), where \(m\) is the slope and \(c\) is the y-intercept. In this case, the slope \(m\) is given as 0.017 and the intercept \(c\) as -5.0. So, the equation becomes \(y = 0.017x - 5\).

Predict gas usage for a specific house size

To predict the gas usage for a 2,100 sq. ft. house, substitute \(x = 2100\) into the regression line equation, we get \(y = 0.017 * 2100 - 5\). Calculating the expression on the right will give the predicted gas usage.

Interpret the meaning of the slope

The slope of the regression line (0.017) indicates the change in gas usage for a unit (1 sq. ft.) increase in size of the house. So, for a 1 sq. ft. increase in size, the gas usage would increase by 0.017 therms.

Determine the limitations of the model

No, the least squares regression line shouldn't be used to predict gas usage for a 500 sq. ft. house. The reason is that it's outside the given range of the data set (1,000 to 3,000 square feet) used to fit the regression line. Using it for predictions outside this range leads to extrapolation, which might not be accurate as the relationship might not hold outside the range of the given data.