1)The director of marketing at
Vanguard Corporation believes the
sales of the company’s Bright Side
Laundry detergent (S) are related to
Vanguard’s own advertising
expenditure (A), as well as the
combined advertising expenditures
of its three biggest rival
detergents R). The marketing
director collects 36 weekly
observations on S, A, and R to
estimate the following multiple
regression equation:
S = a + bA + cR
Where S, A, R are measured in
dollars per week. Vanguard’s
marketing director is comfortable
using parameter estimates that are
statistically significant at the
10 percent level or better.
a) What sign does the marketing
director expect a, b, and c to have?
b) Interpret the coefficients a, b,
and c?
The regression output from the
computer is as follows:
Dependant Variable: S
Observations: 36
R-Square: 0.2247 F-Ratio: 4.781
P-value on F: 0.0150
Variable: Intercept
Parameter Est: 175086.0
Standard Error: 63821.0
T-Ratio: 2.74
P-Value: 0.0098
Variable: A
Paramter estimate: 0.8550
Standard Error: 0.3250
T-Ratio: 2.63
P-Value: 0.0128
Variable: R
Parameter Est: - 0.284
Standard Err: 0.164
T-ratio: - 1.73
P-Value: 0.0927
c) Does Vanguard’s advertising
expenditure have a statistical
significant effect on the sales of
Bright Side detergent? Explain,
using appropriate p-value……
d) Does the advertising by its three
largest rivals affect sales of
Bright Side detergent in a
statistical significant way?
Explain using the appropriate
p-value…….
e) What fraction of the total
variation in sales of Bright Side
remains unexplained?
What can the marketing director do
to increase the explanatory power
of the sales equation?
What other explanatory variables
might be added to this equation?
f) What is the expected level of sales
each week when Vanguard spends
$40,000 per week and the combined
advertising expenditures for the
three rivals are $100,000 per week?
a) The marketing director expects the parameter a to have a positive sign, indicating that there is a baseline level of sales even in the absence of advertising expenditure. The parameter b is expected to have a positive sign, suggesting that an increase in Vanguard's advertising expenditure will lead to an increase in sales. The parameter c is expected to have a negative sign, indicating that an increase in the combined advertising expenditures of the three biggest rival detergents will lead to a decrease in sales of Bright Side detergent.
b) The coefficient a represents the intercept, which indicates the baseline level of sales when both Vanguard's and its rivals' advertising expenditures are zero. The coefficient b represents the effect of Vanguard's advertising expenditure on sales, indicating how much sales increase for each dollar increase in advertising expenditure by Vanguard. The coefficient c represents the effect of the combined advertising expenditures of the three biggest rival detergents on sales, indicating how much sales decrease for each dollar increase in their advertising expenditure.
c) To determine if Vanguard's advertising expenditure has a statistically significant effect on the sales of Bright Side detergent, we need to examine the p-value associated with the coefficient b. The p-value for b is 0.0128, which is less than the significance level of 0.10. Therefore, Vanguard's advertising expenditure has a statistically significant effect on the sales of Bright Side detergent.
d) To determine if the advertising by the three largest rivals affects the sales of Bright Side detergent in a statistically significant way, we need to examine the p-value associated with the coefficient c. The p-value for c is 0.0927, which is greater than the significance level of 0.10. Therefore, the advertising by the three largest rivals does not have a statistically significant effect on the sales of Bright Side detergent.
e) The fraction of the total variation in sales of Bright Side detergent that remains unexplained is given by 1 - R-squared. From the regression output, we can see that the R-squared value is 0.2247. Therefore, the fraction of the total variation in sales that remains unexplained is 1 - 0.2247 = 0.7753, or 77.53%.
To increase the explanatory power of the sales equation, the marketing director can consider adding additional explanatory variables that may affect sales. These variables could include factors such as price, promotions, customer demographics, or economic indicators that may influence consumer behavior and buying decisions.
f) To calculate the expected level of sales each week when Vanguard spends $40,000 per week and the combined advertising expenditures for the three rivals are $100,000 per week, we plug these values into the sales equation.
S = a + bA + cR
S = 175086.0 + 0.8550 * 40000 + (-0.284) * 100000
S = 175086.0 + 34200 + (-28400)
S = 173886
Therefore, the expected level of sales each week in this scenario is $173,886.