Chapter 11.2, Problem 23E

a.

To determine

Find the least-squares regression line to predict the weights from heights.

Answer to Problem 23E

The least-squares regression line to predict the weights from heights is $\underset{\_}{\widehat{y}=107.3+1.59x}$.

Explanation of Solution

Calculation:

The heights (inches) and weights (pounds) for some quarterbacks at a Combine are given.

Denote the heights as x and the weights as y.

Least-squares regression:

For an ordered pairs of values of variables, (x, y) with respective means $\overline{x}$ and $\overline{y}$, respective standard deviations $s_x$ and $s_y$, correlation coefficient r, the least-squares regression line for predicting the response variable, y, by using the predictor variable, x, is given as:

$\widehat{y}=b_0+b_{1}x$, where the slope is $b_1=r\frac{s_y}{s_x}$  and the y-intercept is $b_0=\overline{y}-b_1\overline{x}$.

Regression:

Software procedure:

Step by step procedure to obtain regression using Minitab software is given as,

• Choose Stat > Regression > Regression > Fit Regression Model.
• In Responses, enter the numeric column containing the response data y.
• In Continuous Predictors, enter the numeric column containing the predictor variable x.
• Choose Results, select Regression equation and click OK.
• Click OK.

Output using MINITAB software is given below:

From the output, the least-squares regression line for the data set is found to be $\underset{\_}{\widehat{y}=107.3+1.59x}$.

b.

To determine

Explain whether it is possible to give an interpretation of the y-intercept of the regression line.

Answer to Problem 23E

It is not possible to give an interpretation of the y-intercept of the regression line.

Explanation of Solution

Calculation:

Comparing the least-squares regression equation, $\widehat{y}=107.3+1.59x$ with the general form of the regression equation, the y-intercept is $b_0=107.3$.

The y-intercept would mean that when the height of a quarterback is $x=0$, the weight would be 107.3.

However, it is practically impossible for the height of a quarterback to be 0 inches. The height would always take some non-zero positive value.

Hence, it is not possible to give an interpretation of the y-intercept of the regression line.

c.

To determine

Find the predicted difference in the weights of two quarterbacks, if their heights differ by 2 inches.

Answer to Problem 23E

The predicted difference in the weights of two quarterbacks, if their heights differ by 2 inches is 3.18 pounds.

Explanation of Solution

Interpretation:

It is known that when the predictor variable values differ by the amount d, the corresponding response variable values differ by amount $b_1\cdot d$, where $b_1$ is the slope of the least-squares regression line.

Here, $d=2$.

From part a, the least-squares regression equation is:

$\widehat{y}=107.3+1.59x$.

Comparing this equation with the general form of the regression equation, $b_1=1.59$.

Thus,

$\begin{array}{c}{b}_1\cdot d=2\times 1.59\\ =\mathrm{3.18.}\end{array}$

Hence, the predicted difference in the weights of two quarterbacks, if their heights differ by 2 inches is 3.18 pounds.

d.

To determine

Find the weight of a quarterback whose height is 74.5 inches.

Answer to Problem 23E

The weight of a quarterback whose height is 74.5 inches is 225.755 pounds.

Explanation of Solution

Calculation:

From part a, the least-squares regression equation is:

$\widehat{y}=107.3+1.59x$.

For a quarterback whose height is 74.5 inches, $x=74.5$. Replace this value of x in the equation:

$\begin{array}{c}\widehat{y}=107.3+\left(1.59\times 74.5\right)\\ =107.3+118.455\\ =\mathrm{225.755.}\end{array}$

Hence, the weight of a quarterback whose height is 74.5 inches is 225.755 pounds.

e.

To determine

Find whether the actual weight of Geno Smith is more or less than his predicted weight.

Answer to Problem 23E

The actual weight of Geno Smith is less than his predicted weight.

Explanation of Solution

Calculation:

The actual height of Geno Smith is 74 inches and his actual weight is 218 pounds.

For Geno Smith, height is 74 inches, that is, $x=74$. Replace this value of x in the equation:

$\begin{array}{c}\widehat{y}=107.3+\left(1.59\times 74\right)\\ =107.3+117.66\\ =\mathrm{224.96.}\end{array}$

It is observed that the predicted weight of Geno Smith is 224.96 pounds, which is greater than his actual weight of 218 pounds.

Hence, the actual weight of Geno Smith is less than his predicted weight.