Question

a random sample of records of home sales from feb. 15 to apr. 30, 1993, from the files maintained by the albuquerque board of realtors gives the price and size (in square feet) of 117 hon regression to predict price (in thousands of dollars) from size has an ( r^{2} ) of 71.4%. the residuals plot indicated that a linear model is appropriate. complete parts a through c below.
a) what are the variables and units in this regression?
a. price (in dollars) is y and size (in square feet) is x.
b. price (in dollars) is y and number of houses is x.
c. price (in thousands of dollars) is y and size (in square feet) is x.
d. size (in square feet) is y and price (in thousands of dollars) is x.
b) what units does the slope have?
a. the slope has units of dollars per square foot.
b. the slope has units of thousands of dollars per square foot.
c. the slope has units of dollar amount.
d. the slope has units of square feet per thousands of dollars.

Explanation:

• Part a:
• The regression is to predict Price (in thousands of dollars) from Size (in square feet). In a regression model \(y = a+bx\), \(y\) is the response variable (what we are predicting) and \(x\) is the explanatory variable (used for prediction). So Price (in thousands of dollars) is \(y\) and Size (in square feet) is \(x\).
• Part b:
• The formula for the slope \(b\) in a regression \(y=a + bx\) is \(b=\frac{\sum(x_i-\bar{x})(y_i - \bar{y})}{\sum(x_i-\bar{x})^2}\). The units of \(y\) (Price) are thousands of dollars and units of \(x\) (Size) are square feet. So the units of the slope \(b\) are \(\frac{\text{units of }y}{\text{units of }x}=\frac{\text{thousands of dollars}}{\text{square feet}}\) (thousands of dollars per square foot).

Answer:

a) C. Price (in thousands of dollars) is \(y\) and Size (in square feet) is \(x\)
b) B. The slope has units of thousands of dollars per square foot