Question

ch 5* an owner of a home in the midwest installed solar panels to reduce heating costs. after installing the solar panels, he measured the amount of natural gas used y (in cubic feet) to heat the home and outside temperature x (in degree - days, where a days degree - days are the number of degrees its average temperature falls below 65°f) over a 23 - month period. he then computed the least - squares regression line for predicting y from x and found it to be y^=85 + 16x. the software used to compute the least - squares regression line above says that r² = 0.98. this suggests that prediction of gas used from degree - days will be quite accurate. although degree - days and gas used are correlated, degree - days do not predict gas used very accurately. gas used increases by = 0.99 cubic feet for each additional degree - day.

Explanation:

The coefficient of determination \(r^{2}\) measures the proportion of the variance in the dependent variable (gas used \(y\)) that is predictable from the independent variable (degree - days \(x\)). A value of \(r^{2}=0.98\) (or 98%) means that 98% of the variation in the amount of natural gas used can be explained by the outside temperature (degree - days). This high value of \(r^{2}\) indicates a strong linear relationship, and thus predictions of gas used from degree - days will be quite accurate.

For the second option, since \(r^{2}=0.98\) is very high, it is not the case that degree - days do not predict gas used very accurately.

For the third option, the slope of the regression line \(\hat{y}=85 + 16x\) is 16. The slope represents the change in \(y\) (gas used) for a one - unit change in \(x\) (degree - days). So, gas used increases by 16 cubic feet for each additional degree - day, not 0.99.

Answer:

prediction of gas used from degree - days will be quite accurate.