Question

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. 17 The least-squares line for predicting final-exam score from midterm-exam score was $\overbrace{y}=46.6+0.41x$ . Octavio scores 10 points above the class mean on the midterm. How many points above the class mean do you predict that he will score on the final? (This is an example of the phenomenon that gave “regression” its name: students who do well on the midterm will on the average do less well, but still above average, on the final.)

Short answer

$4.1$points above the class mean he will score on the final

Step-by-step solution

Given Information

Given in the question that Gary Smith of Pomona College looked at the exam scores of all $346$students who took his statistics class over a $10$-year period. $17$The least-squares line for predicting final-exam score from midterm-exam score was $\overbrace{y}=46.6+0.41x$. Octavio scores $10$points above the class mean on the midterm. we have to find out that How many points above the class mean do you predict that he will score on the final.

Explanation

The least-squares line for predicting final-exam score from midterm-exam score was

$\overbrace{Y}=46.6+0.41X$

Octavio scores $10$points above the class mean on the midterm,

That is

$X=\overline{X}+10$

Predicted final score exam is

localid="1650001419325" $$\begin{gathered} \overbrace{Y}=46.6+0.4(\overline{X}+10) \\ =46.6+0.4\overline{X}+0.4\times 10 \\ =(46.6+0.41\overline{X})+4.1 \end{gathered}$$

We use the fact that the least-squares line passes through$(\overline{X},\overline{Y})$:

localid="1649411401399" $\overline{Y}=46.6+0.41\overline{X}$

$\overbrace{Y}=\overline{Y}+4.1$