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 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.
c) do you think the slope is positive or negative? explain.
a. the slope is positive. the price of a home cannot be negative.
b. the slope is negative. as the size of the home increases, the price should decrease.
c. the slope is negative. as the size of the home decreases, the price should also decrease.
d. the slope is positive. as the size of the home increases, the price should also increase.

Explanation:

Part a)

• Identify variables: The problem states "regression to predict Price (in thousands of dollars) from Size". In regression, the variable being predicted is \(y\) (dependent variable) and the variable used for prediction is \(x\) (independent variable). So, Price (in thousands of dollars) is \(y\) and Size (in square feet) is \(x\).

Part b)

• Units of slope: The formula for slope \(b_1=\frac{\sum(x_i - \bar{x})(y_i-\bar{y})}{\sum(x_i - \bar{x})^2}\). The units of \(y\) (Price) are thousands of dollars and units of \(x\) (Size) are square - feet. So, the units of slope are \(\frac{\text{thousands of dollars}}{\text{square feet}}\) (thousands of dollars per square foot).

Part c)

• Sign of slope: In general, as the size of a home (in square feet) increases, the price (in thousands of dollars) of the home also increases. So, there is a positive association between size and price, which means the slope is positive.

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.
c) D. The slope is positive. As the size of the home increases, the price should also increase.