IQ and grades Exercise 3 (page 158) included the plot shown below of school grade point average (GPA) against IQ test score for $78$seventh-grade students. (GPA was recorded on a $12$-point scale with $A+=12,A=11,A-=10,B+=9,...D-=1,F=0$.) Calculation shows that the mean and standard deviation of the IQ scores are$\overline{X}=108.9$and $Sx=13.17$For the GPAs, these values are $\overline{Y}=7.447andSy=2.10$. The correlation between IQ and GPA is $0.6337$

(a) Find the equation of the least-squares line for predicting GPA from IQ. Show your work.

(b) What percent of the observed variation in these students’ GPAs can be explained by the linear relationship between GPA and IQ?

(c) One student has an IQ of $103$but a very low GPA of $0.53$. Find and interpret the residual for this student

Given In the question that We think that " $IQ$scores" will help explain "Grade point average". So $IQscore$is the explanatory variable and "Grade point average" is the response variable. Mean and standard deviation of these variables are given in the table.

localid="1649947222967" $$\begin{gathered} r=0.6337 \\ b=r\frac{Sy}{Sx} \\ a=\overline{Y}-b\overline{X} \end{gathered}$$

Using the least-squares line as a predictor of GPA from IQ, we must determine the equation.

According to a least-squares regression between IQ and GPA, its slope is:

$$\begin{gathered} b=0.06337\times \frac{2.10}{13.17} \\ =0.1010 \end{gathered}$$

Since the least-squares line passes through$(\overline{X},\overline{Y)}$we use this information to find the intercept. :

localid="1649947236918" $$\begin{gathered} a=7.447-0.1010\times 108.9 \\ =-3.5519 \end{gathered}$$

Given in the question that GPA was records on $12$points scale withlocalid="1649437781844" $A+=12,A=11,A-=10,B+=9...D=1,F=0$. We have to find out that percent of the observed variation in these students’ GPAs can be explained by the linear relationship between GPA and IQ.

According to the least-squares regression line of $\mathrm{Y}$on $\mathrm{X}$, the coefficient of determination, $r^2$, represents the percentage of variation in $r$values.

Here,

$$\begin{gathered} r=0.6337 \\ {r}^2=0.4016 \end{gathered}$$

About $40.16$percent of the variation in $\mathrm{Y}$can be explained by the straight-line relationship between $y$and $X.$

$59.84$percent of the variance can't be explained by a linear relationship, which can be explained by individual variance.

Given in the question that One student has an IQ of $103$but a very low GPA of $0.53$we have to find and interpret the residual for this student

According to a least-squares regression between IQ and GPA, its slope is:

$\overbrace{Y}=-3.5519+0.1010X$

Calculating a student's GPA based on an IQ score$103$is

localid="1649947409386" $$\begin{gathered} \left(predicted\right)\overbrace{y}=-3.5519+0.1010\left(103\right) \\ =6.8511 \end{gathered}$$

However, for same student, observed GPA value is

localid="1649947416993" $$\begin{gathered} Y\left(observed\right)=0.53 \\ residual=observedY-predictedY \\ =Y-\hat{Y} \\ =0.53-6.8511 \\ =-6.3211 \end{gathered}$$

There is a negative residual for this data point, which shows below the line