Chapter 3.2, Problem 51E

(a)

To determine

To find: the equation of the least-squares line

Answer to Problem 51E

The equation of the least-squares line is $\widehat{Y}=-3.5519+0.1010X$

Explanation of Solution

Given:

	Mean	Standard deviation
IQ score $\left(X\right)$	$\overline{X}=108.9$	$S_Y=2.10$
Grade point average(GPA) $\left(Y\right)$	$\overline{Y}=7.447$	$S_Y=2.10$

  $r=0.6337$

Calculation:

The least-squares regression line is$\widehat{Y}=a+bX$

Where

Slope $b=r\frac{S_Y}{S_X}$

AndY intercept is $a=\overline{Y}-b\overline{X}$

The least-squares regression line of the IQ score and GPA score has slope:

  $\begin{array}{l}b=r\frac{S_Y}{S_X}\\ b=0.6337\times \frac{2.10}{13.17}\\ =0.1010\end{array}$

To find the intercept, we use the fact that the least-squares line passes through $\left(\overline{X},\overline{Y}\right)$ :

  $\begin{array}{l}a=\overline{Y}-b\overline{X}\\ =7.447-0.1010\times 108.9\\ =-3.5519\end{array}$

So the equation of the least-squares line is:

  $\widehat{Y}=-3.5519+0.1010X$

Conclusion:

So the equation of the least-squares line is $\widehat{Y}=-3.5519+0.1010X$

(b)

To determine

To explain:percent of the variation in these students GPAs

Answer to Problem 51E

Then 40.16% of the variation in Y is due to the straight line relationship between Y and X. The other 59.84% is individual variation that is not explained by the linear relationship.

Explanation of Solution

Calculation:

The coefficient of determination $r^2$ is the fraction of the variation in the values of Y that is accounted for by the least-squares regression line Y on X.

Here,$\begin{array}{l}r=0.6337\\ r^2={\left(0.6337\right)}^2\\ =0.4016\end{array}$

Then about 40.16% of the variation in Y is due to the straight line relationship between Y and X. The other 59.84% is individual variation that is not explained by the linear relationship.

Conclusion:

Hence, about 40.16% of the variation in Y is due to the straight line relationship between Y and X. The other 59.84% is individual variation that is not explained by the linear relationship.

(b)

To determine

To find and interpret: the residual for the students.

Answer to Problem 51E

The student is negative; this shows the data point lies below the line.

The student had a GPA that was 6.3211 points worse than expected for someone with an IQ of 103.

Explanation of Solution

Calculation:

The least-squares regression line of the IQ score and GPA score is:

  $\widehat{Y}=-3.5519+0.1010X$

Predicted GPA score for a student having IQ score 103 is$\begin{array}{l}(Predicted)\widehat{Y}=-3.5519+0.1010\left(103\right)\\ =6.8511\end{array}$

But observed value of GPA score for same student is

  $\begin{array}{l}Y\left(\text{observed}\right)=0.53\\ \text{residual = observed }Y-\text{ predicted }Y\\ =Y-\widehat{Y}\\ =0.53-6.8511\\ =-6.3211\end{array}$

This residual is negative; this shows the data point lies below the line.

The student had a GPA that was 6.3211 points worse than expected for someone with an IQ of 103.

Conclusion:

Hence, the student had a GPA that was -6.3211 points worse than expected for someone with an IQ of 103.