Chapter 12.2, Problem 1.4CYU

To determine

To explain do you expect your prediction to be too high or too low.

Answer to Problem 1.4CYU

Our predictions will be too low.

Explanation of Solution

Since from the previous question we get that,

It is given in the question that variable x be the number of the years since $1700$ and the variable y be the U.S. population in the years $\text{1790 to 1880}$ . Thus, the general equation will be as:

  $\widehat{y}=a+bx$

And the slope and the constant of the regression line is give in the previous question as:

  $\begin{array}{l}a=-1.19311\\ b=0.0287280\end{array}$

Thus, the regression line is as:

  $\begin{array}{l}\widehat{y}=a+bx\\ \Rightarrow \widehat{y}=-1.19311+0.0287280x\end{array}$

Thus, evaluating the regression line we will get that,

  $\begin{array}{l}\widehat{y}=a+bx\\ \Rightarrow \widehat{y}=-1.19311+0.0287280x\\ \Rightarrow \widehat{y}=-1.19311+0.0287280\times 1890\\ \Rightarrow \widehat{y}=53.10281\end{array}$

Thus, based on the residual plot given in this question we can say that the points are first increasing and then it starts decreasing in the year ahead. Thus, in the year $1890$ , we expected the prediction to be too low as the last point in the residual plot is too low so is our prediction will be is if the same trend follows.