Chapter 4.6, Problem 10PPS

a.

To determine

To find: An equation of the regression line.

Answer to Problem 10PPS

The equation of regression line is $y=9.679x+29.071$ .

Explanation of Solution

Given Information: A table showing the prices of the eight top-selling brands of jeans.

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.

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

$x$	$y$	$xy$	$x^2$
1	43	43	1
2	44	88	4
3	50	150	9
4	61	244	16
5	64	320	25
6	135	810	36
7	106	742	49
8	78	624	64

Sum of every column is,

  $\begin{array}{c}{\displaystyle \sum x}=36\\ {\displaystyle \sum y}=581\\ {\displaystyle \sum x}y=3021\\ {\displaystyle \sum x^2}=204\end{array}$

Calculate a as,

  $\begin{array}{c}a=\frac{{\displaystyle \sum y\cdot {{\displaystyle \sum x}}^2}-{\displaystyle \sum x\cdot {\displaystyle \sum x}y}}{n\cdot {\displaystyle \sum x^2}-{\left({\displaystyle \sum x}\right)}^2}\\ =\frac{581\times 204-36\times 3021}{8\times 204-{36}^2}\\ =29.071\end{array}$

Calculate b as,

  $\begin{array}{c}b=\frac{n\cdot {\displaystyle \sum xy}-{\displaystyle \sum x\cdot {\displaystyle \sum y}}}{n\cdot {\displaystyle \sum x^2}-{\left({\displaystyle \sum x}\right)}^2}\\ =\frac{8\times 3021-36\times 581}{8\times 204-{36}^2}\\ =9.679\end{array}$

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

  $\begin{array}{c}y=a+bx\\ y=9.679x+29.071\end{array}$

Hence, the equation of regression line is,

  $y=9.679x+29.071$

b.

To determine

To calculate: The price of a pair of the 12^th best-selling brand.

Answer to Problem 10PPS

The price of a pair of the 12^th best-selling brand is 145.219

Explanation of Solution

Given Information: The equation of regression line is $y=9.679x+29.071$ .

Calculation:

Substitute $x=12$ in the equation of regression line $y=9.679x+29.071$ ,

  $\begin{array}{c}y=9.679x+29.071\\ =9.679\left(12\right)+29.071\\ =145.219\end{array}$

The price of a pair of the 12^th best-selling brand is 145.219.

c.

To determine

To find: Whether the prediction is reasonable.

Answer to Problem 10PPS

No, the prediction is not reasonable.

Explanation of Solution

Given Information: The price of a pair of the 12^th best-selling brand is 145.219.

Calculation:

From the table it is evident that the prices of the top-6 selling brands are low but slowly increasing. The second pattern is that for the 7^th and 8^th top-selling brands, the prices are decreasing. So, if this pattern continues, the price will decrease and since our prediction is an increased value it is not a reasonable prediction.