Chapter 1.4, Problem 27E

a.

To determine

To find: linear equation that can be used to identify car’s highway mileage based on city mileage.

Answer to Problem 27E

  $y=1.35x\pm 5.55$

Explanation of Solution

Given information:

The given table is as follows:

Model	City (mpg)	Highway (mpg)
A	24	32
B	20	29
C	20	29
D	20	28
E	23	30
F	24	30
G	27	37
H	22	28

Calculation:

There are total 8 models of car from A to H.

Let x be the city mileage and y be the highway mileage. The relation of the given data will be as follows:

  $\left\{\left(24,32\right),\left(20,29\right),\left(20,29\right),\left(20,28\right),\left(23,30\right),\left(24,30\right),\left(27,37\right),\left(22,28\right)\right\}$

Evaluate the average domain value ( x ) as follows:

  $\begin{array}{c}\text{Average domain}=\frac{24+20+20+20+23+24+27+22}{8}\\ =\frac{180}{8}\\ =22.5\end{array}$

Thus, the value of x is 22.5.

Evaluate the average range value ( y ) as follows:

  $\begin{array}{c}\text{Average range}=\frac{32+29+29+28+30+30+37+28}{8}\\ =\frac{243}{8}\\ =30.375\end{array}$

Thus, the value of y is 30.375.

Put the value of x and y in the slope-intercept form and evaluate m as follows:

  $\begin{array}{c}y=mx+b\\ 30.375=m\left(22.5\right)+b\\ 30.375=22.5m+0\left[\text{Assume }b=0\right]\\ m=\frac{30.375}{22.5}\\ =1.35\end{array}$

Put $m=1.35$ and $b=0$ in the slope intercept equation as follows:

  $y=1.35x$  ... (1)

Now put the value of x in equation (1) to evaluate the new value of y (let’s say, $y_1$ ) and then find the error. The table below shows the values of margin error in each case.

Model	$x$	$y$	$y_1$	$\text{Error}=y-y_1$
A	24	32	32.4	+0.4
B	20	29	27	$-2$
C	20	29	27	$-2$
D	20	28	27	$-1$
E	23	30	31.05	+1.05
F	24	30	32.4	+2.4
G	27	37	36.45	$-0.55$
H	22	28	29.7	+1.7

Now, evaluate the sum of positive errors and negative errors as shown below.

  $\begin{array}{c}\text{Positive errors}=+0.4+1.05+2.4+1.7\\ =+5.55\\ \text{Negative errors}=-2-2-1-0.55\\ =-5.55\end{array}$

Thus, the margin of error is $\pm 5.55$ .

Therefore, the equation is $y=1.35x\pm 5.55$ .

b.

To determine

To predict: highway mileage of J’s city using the equation.

Answer to Problem 27E

  $25.65\text{mpg}\pm 5.55$

Explanation of Solution

Given information:

The given table is as follows:

Model	City (mpg)	Highway (mpg)
A	24	32
B	20	29
C	20	29
D	20	28
E	23	30
F	24	30
G	27	37
H	22	28

City mileage of Model J =19mpg

The equation representing highway mileage is as follows;

  $y=1.35x\pm 5.55$

Here x is city mileage and y is highway mileage.

Calculation:

Substitute $x=19\text{ mpg}$ in the given equation and evaluate city mileage of model J as shown below.

  $\begin{array}{c}y=1.35\left(19\text{mpg}\right)\pm 5.55\\ =25.65\text{mpg}\pm 5.55\end{array}$

Thus, the highway mileage of model J is $25.65\text{mpg}\pm 5.55$ .

c.

To determine

To explain: the effectiveness of the equation to predict the mileage

Answer to Problem 27E

Since the difference is within the margin of error. Thus, the predicted mileage value is very much accepted.

Explanation of Solution

Given information:

The given table is as follows:

Model	City (mpg)	Highway (mpg)
A	24	32
B	20	29
C	20	29
D	20	28
E	23	30
F	24	30
G	27	37
H	22	28

Highway mileage of Model J =26mpg

The equation representing highway mileage is as follows;

  $y=1.35x\pm 5.55$

Here x is city mileage and y is highway mileage.

The evaluated value y for model J = $25.65\text{mpg}\pm 5.55$

Calculation:

The difference between the evaluated and given value of highway mileage of model J is as follows:

  $\begin{array}{c}\text{Difference}=\text{Given }y-\text{Evaluated }y\\ =26-25.65\\ =0.35\end{array}$

Since the difference is within the margin of error. Thus, the predicted value is very much accepted.