Chapter 3.2, Problem 58E

(a)

To determine

To calculate and interpret the residual for the student who was $141$ cm tall at age $10$ .

Answer to Problem 58E

The residual is $-7.2$ .

Explanation of Solution

The relationship between the height and age of the students is given in the question. And the regression line is as:

  $\widehat{y}=106.1+4.21x$

And also given that the student who was $141$ cm tall at age $10$ . Thus, we have,

  $\begin{array}{l}x=10\\ y=141\end{array}$

Thus the predicted height of the student is as:

  $\begin{array}{l}\widehat{y}=106.1+4.21x\\ =106.1+4.21(10)\\ =148.2\end{array}$

Thus, the residual will be calculated as:

  $\begin{array}{l}\text{Residual}=y-\widehat{y}\\ =141-148.2\\ =-7.2\end{array}$

This then implies that we overestimated the height of the student at age $10$ by $7.2$ cm, when making a prediction using the regression line.

(b)

To determine

To interpret the slope of the least square regression line.

Explanation of Solution

The relationship between the height and age of the students is given in the question. And the regression line is as:

  $\widehat{y}=106.1+4.21x$

And also given that the student who was $141$ cm tall at age $10$ . Thus, we have,

  $\begin{array}{l}x=10\\ y=141\end{array}$

As we know that the slope is coefficient of x in the least square regression equation and represents the average increase or decrease of y per unit of x . Thus,

  $b_1=4.21$

This then implies that on average, the height increases by $4.21$ cm per year.

(c)

To determine

To interpret the value of s .

Explanation of Solution

The relationship between the height and age of the students is given in the question. And the regression line is as:

  $\widehat{y}=106.1+4.21x$

And also given that the student who was $141$ cm tall at age $10$ . Thus, we have,

  $\begin{array}{l}x=10\\ y=141\end{array}$

And the value of s is, $s=8.61$ . As we know that the standard error of the estimate s represents the average error of predictions thus the average deviation between actual and the predicted values. Thus the predicted height of a student using the equation of least square regression line deviates on average by $8.61$ cm from the actual height of the student.

(d)

To determine

To interpret the value of $r^2$ .

Explanation of Solution

The relationship between the height and age of the students is given in the question. And the regression line is as:

  $\widehat{y}=106.1+4.21x$

And also given that the student who was $141$ cm tall at age $10$ . Thus, we have,

  $\begin{array}{l}x=10\\ y=141\end{array}$

And the value of $r^2$ is,

  $\begin{array}{l}{r}^2=0.274\\ =27.4\%\end{array}$

As we know that the coefficient of determination measures the proportion of variation in the responses y variable that is explained by the least square regression model using the explanatory x variable. Thus, we can say that $27.4\%$ of the variation in the height is explained by the least square regression line using the age as explanatory variable.