Will I bomb the final? We expect that students who do well on the midterm exam in a course will usually also do well on the final exam. Gary Smith of Pomona College looked at the exam scores of all 346 students who took his statistics class over a 10-year period. Assume that both the midterm and final exams were scored out of 100 points.

a. State the equation of the least-squares regression line if each student scored the same on the midterm and the final.

b. The actual least-squares line for predicting final exam score y from midterm-exam score x was y^=46.6+0.41x. Predict the score of a student who scored

50 on the midterm and a student who scored 100 on the midterm.

c. Explain how your answers to part (b) illustrate regression to the mean.

Over a ten-year period, Gary Smith of Pomona College examined the exam scores of all $346$ students who attended his statistics class. Assume that the midterm and final exams were both graded on a scale of $100$

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

If each student gets the same grade on the midterm and final, the data for the explanatory variable $x$ and the response variable y will be the same. If the data for the two variables is the same, the linear model will forecast that they are equal, and so the linear model will presume that the expected $y-$variable is equal to the predicted $x-$variable. As a result, the least-square regression line looks like this:

$\stackrel{⏜}{y}=x$

The least-square regression line is given as follows in the question:

$\stackrel{⏜}{y}=46.6+0.41x$

To forecast the grade of a student who had a $50$ on the midterm, we must use the following formula:

$$\begin{gathered} \stackrel{⏜}{y}=46.6+0.41x \\ =46.6+0.41\left(50\right) \\ =67.10 \end{gathered}$$

As a result, a student who had a $50$ on the midterm will have a predicted score of $67.10$

Now, in order to forecast the grade of a student who had a perfect score on the midterm, we must use the following formula:

$$\begin{gathered} \stackrel{⏜}{y}=46.6+0.41x \\ =46.6+0.41\left(100\right) \\ =87.60 \end{gathered}$$

Thus the predicted score of a student who scored $100$on the midterm is $87.10$

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 $87.60$

As a result, we anticipate that the midterm's mean will be between $50$ and $100$ assuming that the majority of students pass the midterm. Similarly, the same applies to the final score. The student who received a $50$ on the midterm then received a grade that was lower than the mean. The student who received a perfect score on the midterm then outperformed the mean. Finally, we note that the student who received a $50$ has a higher anticipated final score, whereas the student who received a $100$ has a lower predicted final score, indicating that the final exam prediction appears to be closer to the mean.