Chapter 3.2, Problem 65E

(a)

To determine

To state the equation of the least squares regression line if each student scored the same on the midterm and final.

Answer to Problem 65E

  $\widehat{y}=x$ .

Explanation of Solution

Let x be midterm score and y be the final score.

If each student scores the same on the midterm and the final then the data of the explanatory variable x will be the same as the data for the response variable y . If the data of the two variables are the same then the linear model will predict that the two variables are equal and thus the linear model will assume that the predicted y -variable is equal to the x -variable. Thus, we have the least square regression line as:

  $\widehat{y}=x$ .

(b)

To determine

To predict the score of a student who scored $50$ on the midterm and a student who scored $100$ on the midterm.

Answer to Problem 65E

The predicted score of a student who scored $50$ on the midterm is $67.10$ .

The predicted score of a student who scored $100$ on the midterm is $87.60$ .

Explanation of Solution

It is given in the question the least square regression line be as:

  $\widehat{y}=46.6+0.41x$

To predict the score of a student who scored $50$ on the midterm, then we have,

  $\begin{array}{l}\widehat{y}=46.6+0.41x\\ =46.6+0.41(50)\\ =67.10\end{array}$

Thus the predicted score of a student who scored $50$ on the midterm is $67.10$ .

Now, to predict the score of a student who scored $100$ on the midterm, we have,

  $\begin{array}{l}\widehat{y}=46.6+0.41x\\ =46.6+0.41(100)\\ =87.60\end{array}$

Thus the predicted score of a student who scored $100$ on the midterm is $87.60$ .

(c)

To determine

To explain how your answer to part (b) illustrate regression to the mean.

Explanation of Solution

We have from part (b) as:

The predicted score of a student who scored $50$ on the midterm is $67.10$ .

The predicted score of a student who scored $100$ on the midterm is $87.60$ .

Thus, we estimate that the mean of the midterm will be somewhere between $50$ and $100$ ,assuming that most students pass the midterm. Similarly, this is for the score of the final. The student who score $50$ on the midterm then scored lower that the mean. And the student who scored $100$ on the midterm then scored higher than the mean. Finally, we also note that the student who scored $50$ has the higher predicted score on the final, while the student who scored $100$ has a lower predicted score on the final, which indicates that the prediction on the final exam appears to be closer to the mean.