Chapter 4.6, Problem 8PPS

a.

To determine

To find: An equation of the regression line.

Answer to Problem 8PPS

The equation of regression line is $y=3.067x+32.711$ .

Explanation of Solution

Given Information: A table showing the record of the number of campsites rented.

X is the number since 2007

Calculation:

An equation of the best fit line is given by,

  $y=a+bx$ …… (1)

• Here, a is the intercept and b is the slope.

The years start from 2009 and go on till 2017.

Assume the year 2007 as 0 and x be the number of years since 2007.

Find $xy\text{ and }{x}^2$ ,

$x$	$y$	$xy$	$x^2$
2	34	68	4
3	45	135	9
4	42	168	16
5	53	265	25
6	58	348	36
7	47	329	49
8	57	456	64
9	65	585	81
10	59	590	100

Sum of every column is,

  $\begin{array}{c}{\displaystyle \sum x}=54\\ {\displaystyle \sum y}=460\\ {\displaystyle \sum x}y=2944\\ {\displaystyle \sum x^2}=384\end{array}$

Calculate a as,

  $\begin{array}{c}a=\frac{{\displaystyle \sum y\cdot {{\displaystyle \sum x}}^2}-{\displaystyle \sum x\cdot {\displaystyle \sum x}y}}{n\cdot {\displaystyle \sum x^2}-{\left({\displaystyle \sum x}\right)}^2}\\ =\frac{460\times 384-54\times 2944}{9\times 384-{54}^2}\\ =32.711\end{array}$

Calculate b as,

  $\begin{array}{c}b=\frac{n\cdot {\displaystyle \sum xy}-{\displaystyle \sum x\cdot {\displaystyle \sum y}}}{n\cdot {\displaystyle \sum x^2}-{\left({\displaystyle \sum x}\right)}^2}\\ =\frac{9\times 2944-54\times 460}{9\times 384-{54}^2}\\ =3.067\end{array}$

Substitute the values of a and b in equation (1),

  $\begin{array}{c}y=a+bx\\ y=3.067x+32.711\end{array}$

• Here, x be the number of years since 2007.

Hence, the equation of regression line is,

  $y=3.067x+32.711$

b.

To determine

To calculate: The number of campsites that will be rented in 2019.

Answer to Problem 8PPS

The number of campsites that will be rented in 2019 is 69.515.

Explanation of Solution

Given Information: The equation of regression line is $y=3.067x+32.711$ .

Calculation:

Substitute $x=12$ in the equation of regression line is $y=3.067x+32.711$ , as 2019 is 12 years after 2007.

  $\begin{array}{c}y=3.067x+32.711\\ =3.067\left(12\right)+32.711\\ =69.515\end{array}$

The number of campsites that will be rented in 2019 is 69.515.

c.

To determine

To calculate: The number of campsites that will be rented in 2027.

Answer to Problem 8PPS

The number of campsites that will be rented in 2027 is 94.051

Explanation of Solution

Given Information: The equation of regression line is $y=3.067x+32.711$ .

Calculation:

Substitute $x=20$ in the equation of regression line is $y=3.067x+32.711$ , as 2027 is 20 years after 2007.

  $\begin{array}{c}y=3.067x+32.711\\ =3.067\left(20\right)+32.711\\ =94.051\end{array}$