Chapter 3.2, Problem 20AYU

a.

To determine

Use a graphing utility to draw scatter diagram.

Answer to Problem 20AYU

  

Explanation of Solution

Given information:

A paediatrician wanted to find a linear model that relates a child’s height, $H$ , to head circumference, $C$ . She randomly selects nine children from her practice, measures their height and head circumfence and obtains the data shown. Let $H$ represent the independent variable and $C$ the dependent variable.

  $\begin{array}{cccccccccc}\begin{array}{l}HeightH\\ \left(inches\right)\end{array}& 25.25& 25.75& 25& 27.75& 26.5& 27& 26.75& 26.75& 27.5\\ \begin{array}{l}HeadCircumference,C\\ \left(inches\right)\end{array}& 16.4& 16.9& 16.9& 17.6& 17.3& 17.5& 17.3& 17.5& 17.5\end{array}$

Use a graphing utility to draw scatter diagram.

Calculation:

The scatter diagram is as shown,

  

Hence, The scatter diagram is as plotted.

b.

To determine

Express the model using function notation.

Answer to Problem 20AYU

  $\boxed{C\left(H\right)=0.3734H+7.3268}$

Explanation of Solution

Given information:

A paediatrician wanted to find a linear model that relates a child’s height, $H$ , to head circumference, $C$ . She randomly selects nine children from her practice, measures their height and head circumfence and obtains the data shown. Let $H$ represent the independent variable and $C$ the dependent variable.

  $\begin{array}{cccccccccc}\begin{array}{l}HeightH\\ \left(inches\right)\end{array}& 25.25& 25.75& 25& 27.75& 26.5& 27& 26.75& 26.75& 27.5\\ \begin{array}{l}HeadCircumference,C\\ \left(inches\right)\end{array}& 16.4& 16.9& 16.9& 17.6& 17.3& 17.5& 17.3& 17.5& 17.5\end{array}$

Use a graphing utility to find the line of best fit that models the relation between height and head circumference. Express the model using function notation.

Calculation:

Express the model using function notation.

  $\begin{array}{l}X:Vector\left(25,25,25.75,25,27.75,26.5,27,26.75,26.75,27.5\right)\\ Y:Vector\left(16.4,16.9,16.9,17.6,17.3,17.3,17.5,17.3,17.5,17.5\right)\end{array}$

  $\begin{array}{l}fit\left(a+b,t,X,Y,t\right)\\ 7.32680608365025+0.373384030418250412\end{array}$

We know the linear function,

  $f\left(x\right)=mx+b$

  $C\left(H\right)=0.3734H+7.3268$

Hence, the line in function notation is, $\boxed{C\left(H\right)=0.3734H+7.3268}$

c.

To determine

Interpret the slope.

Answer to Problem 20AYU

Head circumference is increased by smaller amount as compared to the height of the child.

Explanation of Solution

Given information:

A paediatrician wanted to find a linear model that relates a child’s height, $H$ , to head circumference, $C$ . She randomly selects nine children from her practice, measures their height and head circumfence and obtains the data shown. Let $H$ represent the independent variable and $C$ the dependent variable.

  $\begin{array}{cccccccccc}\begin{array}{l}HeightH\\ \left(inches\right)\end{array}& 25.25& 25.75& 25& 27.75& 26.5& 27& 26.75& 26.75& 27.5\\ \begin{array}{l}HeadCircumference,C\\ \left(inches\right)\end{array}& 16.4& 16.9& 16.9& 17.6& 17.3& 17.5& 17.3& 17.5& 17.5\end{array}$

Interpret the slope.

Calculation:

The slope of line $C\left(H\right)=0.3734H+7.3268$ is $\boxed{0.3734}$ .

As the slope of the line is less than one so head circumference is increased by smaller amount as compared to the height of the child.

Hence, head circumference is increased by smaller amount as compared to the height of the child.

d.

To determine

Predict the head circumference of childe that is $26$ inches tall.

Answer to Problem 20AYU

  $\boxed{C=17.0352}$

Explanation of Solution

Given information:

A paediatrician wanted to find a linear model that relates a child’s height, $H$ , to head circumference, $C$ . She randomly selects nine children from her practice, measures their height and head circumfence and obtains the data shown. Let $H$ represent the independent variable and $C$ the dependent variable.

  $\begin{array}{cccccccccc}\begin{array}{l}HeightH\\ \left(inches\right)\end{array}& 25.25& 25.75& 25& 27.75& 26.5& 27& 26.75& 26.75& 27.5\\ \begin{array}{l}HeadCircumference,C\\ \left(inches\right)\end{array}& 16.4& 16.9& 16.9& 17.6& 17.3& 17.5& 17.3& 17.5& 17.5\end{array}$

Predict the head circumference of childe that is $26$ inches tall.

Calculation:

The head circumference of childe that is $26$ inches tall.

Put $H=26$ in equation, $C\left(H\right)=0.3734H+7.3268$ .

  $\begin{array}{l}C=0.3734\times 26+7.3268\\ =9.7084+7.3268\end{array}$

  $C=17.0352$

Hence, the head circumference of childe that is $26$ inches tall, $\boxed{C=17.0352}$

e.

To determine

What is height of child whose head circumference is $17.4inches$ .

Answer to Problem 20AYU

  $\boxed{H=26.98inches}$

Explanation of Solution

Given information:

A paediatrician wanted to find a linear model that relates a child’s height, $H$ , to head circumference, $C$ . She randomly selects nine children from her practice, measures their height and head circumfence and obtains the data shown. Let $H$ represent the independent variable and $C$ the dependent variable.

  $\begin{array}{cccccccccc}\begin{array}{l}HeightH\\ \left(inches\right)\end{array}& 25.25& 25.75& 25& 27.75& 26.5& 27& 26.75& 26.75& 27.5\\ \begin{array}{l}HeadCircumference,C\\ \left(inches\right)\end{array}& 16.4& 16.9& 16.9& 17.6& 17.3& 17.5& 17.3& 17.5& 17.5\end{array}$

What is height of child whose head circumference is $17.4inches$ .

Calculation:

Put $C=17.4$ in equation, $C\left(H\right)=0.3734H+7.3268$ .

  $\begin{array}{l}17.4=0.373H+7.3268\\ 0.373H=10.0732\\ H=26.98inches\end{array}$

  $\boxed{H=26.98inches}$

Hence, the height of child is, $\boxed{H=26.98inches}$