Chapter 3, Problem R3.5RE

(a)

To determine

To find the equation for the regression line Professor Friedman should use to make the prediction.

Answer to Problem R3.5RE

  $\widehat{y}=30.2+0.16x$ .

Explanation of Solution

It is given that:

  $\begin{array}{l}r=0.6\\ \overline{x}=280\\ \overline{y}=75\\ s_x=30\\ s_y=8\end{array}$

Thus the slope and the constant will be calculated as:

  $\begin{array}{l}b=r\frac{s_y}{s_x}\\ =0.6\times \frac{8}{30}\\ =0.16\\ a=\overline{y}-b\overline{x}\\ =75-0.16\times 280\\ =30.2\end{array}$

Thus the regression line will be as:

  $\begin{array}{l}\widehat{y}=a+bx\\ \Rightarrow \widehat{y}=30.2+0.16x\end{array}$

(b)

To determine

To use the regression line to predict Julie’s final exam score.

Answer to Problem R3.5RE

  $78.2$ is Julie’s exam score.

Explanation of Solution

The regression line is:

  $\widehat{y}=30.2+0.16x$

Thus, the Julie’s final exam score be as:

  $\begin{array}{l}\widehat{y}=30.2+0.16x\\ =30.2+0.16(300)\\ =78.2\end{array}$

(c)

To determine

To explain the meaning of the “least-squares” in the context of this question.

Explanation of Solution

The least squares regression line is the line that minimizes the squared residuals. The residuals are the differences between the observed an the predicted values. Thus the least squares regression line is then the line that minimizes the squared differences between the actual final score and the predicted final exam score.

(d)

To determine

To determine $r^2$ and use this result to argue her actual score could have been much higher than the predicted value.

Answer to Problem R3.5RE

  $\begin{array}{l}{r}^2=0.36\\ =36\%\end{array}$

Explanation of Solution

The value $r^2$ is calculated as:

  $\begin{array}{l}{r}^2={0.6}^2\\ =0.36\\ =36\%\end{array}$

This means that $36\%$ of the variation between the variables have been explained by the regression line. Because only $36\%$ of the variation of the final exam score is explained by the regression line, the final score could deviate strongly from the predicted scores by this regression line.