Question

The following data represent a company’s yearly sales volume and its advertising expenditure over a period of 8 years.

Year                            Sales (Y)                                  Advertising (X)

1989                            15                                                        32

1990                            16                                                        33

1991                            18                                                        35

1992                            17                                                        34

1993                            16                                                        36

1994                            19                                                        37

1995                            19                                                        39

1996                            24                                                        42

1. Develop a scatter diagram of sales versus advertising
2. Use the method of least squares to compute an estimated regression line between sales and advertising.
3. If the company’s advertising expenditure is $40, what is the predicted sales?
4. What does the slope of the estimated regression line indicate? (The relationship btw the variables)
5. Compute the coefficient of determination and interpret its meaning.
6. Compute the correlation coefficient.

Answer 1

a.

b.

Sum of X = 288
Sum of Y = 144
Mean X = 36
Mean Y = 18
Sum of squares (SS_X) = 76
Sum of products (SP) = 60

Regression Equation = ŷ = bX + a

b = SP/SS_X = 60/76 = 0.7895

a = M_Y - bM_X = 18 - (0.79*36) = -10.4211

ŷ = 0.7895X - 10.4211

c. For x=40, ŷ = (0.7895*40) - 10.4211=21.1589

d. As slope value is positive, there is positive relationship between advertising and sales

For every increase in x, y will change 0.7895 correspondingly and it is slope value

e. First we will find r

X Values
∑ = 288
Mean = 36
∑(X - M_x)² = SS_x = 76

Y Values
∑ = 144
Mean = 18
∑(Y - M_y)² = SS_y = 56

X and Y Combined
N = 8
∑(X - M_x)(Y - M_y) = 60

R Calculation
r = ∑((X - M_y)(Y - M_x)) / √((SS_x)(SS_y))

r = 60 / √((76)(56)) = 0.9197

So coefficient of determination is r^2=0.9197^2=0.8458

f. R Calculation
r = ∑((X - M_y)(Y - M_x)) / √((SS_x)(SS_y))

r = 60 / √((76)(56)) = 0.9197