Question

the regression of price on size of homes had r² = 31.6%. answer the questions below.
a) what is the correlation between size and price? r = 0.562 (round to three decimal places as needed.)
b) what would you predict about the price of a home one standard deviation above average in size?
a. the price of a home that is one standard deviation above the mean size would be predicted to cost r times the mean price.
b. the price of a home that is one standard deviation above the mean size would be predicted to be r standard deviations above the mean price.
c. the price of a home that is one standard deviation above the mean size would be predicted to be r standard deviations below the mean price.
d. there is not enough information to make a prediction.

Explanation:

Step1: Recall the formula for correlation coefficient and prediction in regression

The relationship between the standardized variables in a simple - linear regression is given by the correlation coefficient $r$. If $x$ and $y$ are the variables (in this case, size and price of homes), and $z_x$ and $z_y$ are their standardized values ($z=\frac{x - \mu}{\sigma}$), then $z_y=r\times z_x$.

Step2: Solve part (a)

The coefficient of determination $R^{2}=31.6\% = 0.316$. The correlation coefficient $r=\pm\sqrt{R^{2}}$. Since the relationship between size and price is likely positive (larger homes usually cost more), $r=\sqrt{0.316}\approx0.562$.

Step3: Solve part (b)

We know that in a standardized regression model, if $z_x = 1$ (one - standard deviation above the mean in size), then $z_y=r\times z_x$. Given $r = 0.562$ and $z_x = 1$, the standardized price $z_y$ (number of standard deviations of the price from the mean price) is $z_y=r\times1=0.562$. So the price of a home one - standard deviation above the mean size would be predicted to be $r = 0.562$ standard deviations above the mean price.

Answer:

a) $r\approx0.562$
b) B. The price of a home that is one standard deviation above the mean size would be predicted to be $r$ standard deviations above the mean price.