In: Statistics and Probability
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 |
32 |
15 |
33 |
16 |
34 |
18 |
34 |
17 |
35 |
16 |
37 |
18 |
39 |
21 |
40 |
24 |
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. |
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 (SSX) = 58
Sum of products (SP) = 55.5
Regression Equation = ŷ = bX + a
b = SP/SSX = 55.5/58 =
0.9569
a = MY - bMX = 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 - Mx)2 = SSx = 58
Y Values
∑ = 145
Mean = 18.125
∑(Y - My)2 = SSy = 62.875
X and Y Combined
N = 8
∑(X - Mx)(Y - My) = 55.5
R Calculation
r = ∑((X - My)(Y - Mx)) /
√((SSx)(SSy))
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 - Mx)2 = SSx = 58
Y Values
∑ = 145
Mean = 18.125
∑(Y - My)2 = SSy = 62.875
X and Y Combined
N = 8
∑(X - Mx)(Y - My) = 55.5
R Calculation
r = ∑((X - My)(Y - Mx)) /
√((SSx)(SSy))
r = 55.5 / √((58)(62.875)) = 0.9191