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. question 18 1 pts ch 5 fred keeps his savings in his mattress. he began with $1000 from his mother and adds $100 each year. his total savings y after x years are given by the equation y = 1000 + 100x. y = 100x. y = 100 + 1000x. y = 1000 + x.

Explanation:

Step1: Analyze the coefficient of determination ($r^{2}$)

The coefficient of determination $r^{2}$ measures the proportion of the variance in the dependent variable (in the first problem, it's the amount of natural gas used \(y\)) that is predictable from the independent variable (outside temperature \(x\) measured in degree - days). A value of \(r^{2}=0.98\) (or 98%) means that 98% of the variation in \(y\) can be explained by the linear relationship with \(x\). This implies that the regression line \(y^{\wedge}=85 + 16x\) is a very good fit for the data, and predictions of \(y\) from \(x\) will be quite accurate.

Step2: Analyze the savings equation

For the second problem, Fred starts with an initial amount of \(y = 1000\) (when \(x = 0\), this is the y - intercept of the linear equation). He adds \(100\) each year. In a linear equation of the form \(y=mx + b\), where \(m\) is the slope (rate of change) and \(b\) is the y - intercept. Here, the slope \(m = 100\) (the amount he adds per year) and \(b=1000\) (the initial amount). So the equation is \(y = 1000+100x\).

Answer:

For the first part: prediction of gas used from degree - days will be quite accurate.
For the second part: \(y = 1000 + 100x\)