Question

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

Advertising Expenditure (in $10,000) (x)	Sales in Millions of Dollars (y)
32	15
33	16
34	18
34	17
35	16
37	18
39	21
40	24

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a.	Develop a scatter diagram of sales versus advertising and explain what it shows regarding the relationship between sales and advertising.
b.	Compute an estimated regression equation between sales and advertising.
c.	If the company's advertising expenditure is $400,000, what are the predicted sales? Give the answer in dollars.
d.	What does the slope of the estimated regression line indicate in the context of this problem?
e.	Compute the coefficient of determination and fully interpret its meaning in the context of the problem..
f.	Compute the correlation coefficient.

Answer 1

a.

We see that there is increasing trend hence it is positive correlation between x and y

b.

Sum of X = 284
Sum of Y = 145
Mean X = 35.5
Mean Y = 18.125
Sum of squares (SS_X) = 58
Sum of products (SP) = 55.5

Regression Equation = ŷ = bX + a

b = SP/SS_X = 55.5/58 = 0.9569

a = M_Y - bM_X = 18.13 - (0.96*35.5) = -15.8448

ŷ = 0.9569X - 15.8448

c. For x=40, ŷ = (0.9569*40) - 15.8448=22.4312

d. Slope is the rate change in y corresponding to rate change in x

So for single increase in x, y changes to 0.9569

e.

X Values
∑ = 284
Mean = 35.5
∑(X - M_x)² = SS_x = 58

Y Values
∑ = 145
Mean = 18.125
∑(Y - M_y)² = SS_y = 62.875

X and Y Combined
N = 8
∑(X - M_x)(Y - M_y) = 55.5

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

r = 55.5 / √((58)(62.875)) = 0.9191

So r^2=0.8447

So 84.47% of variation in y is explained by x

f. X Values
∑ = 284
Mean = 35.5
∑(X - M_x)² = SS_x = 58

Y Values
∑ = 145
Mean = 18.125
∑(Y - M_y)² = SS_y = 62.875

X and Y Combined
N = 8
∑(X - M_x)(Y - M_y) = 55.5

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

r = 55.5 / √((58)(62.875)) = 0.9191