Chapter 13, Problem 13.26SE

a)

To determine

To find: The equation of the line relating E ( y ) to advertising expenditure $x_1$ for the men’s wear department.

Explanation of Solution

Given:

The data is:

Department	1	2	3	4	5
Men's wear A	5.2	5.9	7.7	7.9	9.4
Children’s wear B	8.2	9	9.1	10.5	10.5
Women's wear C	10	10.3	12.1	12.7	13.6

Calculation:

The Minitab output is:

  

Thus, the least-square line is:

  $\begin{array}{c}E\left(y\right)=4.10+1.04x_1+3.53x_2+4.76x_3-0.430x_1{x}_2-0.080x_1{x}_3\\ =4.10+1.04x_1+3.53\left(0\right)+4.76\left(0\right)-0.43\left(0\right)-0.080\left(0\right)\\ =4.10+1.04x_1\end{array}$

b)

To determine

To find: The equation of the line E ( y ) to x₁ .

Explanation of Solution

The equation of the line E ( y ) to x₁ can be calculated as:

  $\begin{array}{c}E\left(y\right)=4.10+1.04x_1+3.53x_2+4.76x_3-0.430x_1{x}_2-0.080x_1{x}_3\\ =4.10+1.04x_1+3.53\left(1\right)+4.76\left(0\right)-0.43x_1-0.080\left(0\right)\\ =4.10+1.04x_1+3.53-0.430x_1\\ =7.63+0.61x_1\end{array}$

c)

To determine

To find: The equation of the line E ( y ) to x₁ for the women’s wear department C.

Explanation of Solution

The equation of the line E ( y ) to x₁ for the women’s wear department C can be calculated as:

  $\begin{array}{c}E\left(y\right)=4.10+1.04x_1+3.53x_2+4.76x_3-0.430x_1{x}_2-0.080x_1{x}_3\\ =4.10+1.04x_1+3.53\left(0\right)+4.76\left(1\right)-0.43\left(1\right)-0.080x_1\\ =4.10+1.04x_1+4.76-0.080x_1\\ =8.86+0.96x_1\end{array}$

d)

To determine

To find: The difference between the intercepts of the E ( y ) lines corresponding to the children’s wear B and men’s wear A departments.

Explanation of Solution

The difference between the intercepts of the E ( y ) lines corresponding to the children’s wear B and men’s wear A departments can be calculated by taking the difference of above two parts as shown below:

  $E\left(y\right)-4.10-1.04x_1=E\left(y\right)-7.63-0.61x_1$

The difference between the intercept is 3.53.

e)

To determine

To find: The difference between the slopes of the E ( y ) lines corresponding to the children’s wear B and men’s wear A departments.

Explanation of Solution

The difference between the intercepts of the E ( y ) lines corresponding to the children’s wear B and men’s wear A departments can be calculated by taking the difference of above two parts as shown below:

  $E\left(y\right)-4.10-1.04x_1=E\left(y\right)-7.63-0.61x_1$

The difference between the slopes is 0.08.

f)

To determine

To find: Whether the slope of the lines corresponding to the three departments are equal.

Explanation of Solution

The Excel output is:

  

From the Excel output, the p -value is less than the level of significance. The null hypothesis is rejected. Thus, it could be concluded that the slope of lines corresponding to three departments are unequal.