Week 2 Applied, problem 1 The director of marketing at Vanguard Corporation believes that sales of the company’s Bright Side laundry detergent (S) are related to Vanguard’s own advertising expenditures (A), as well as 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, and R, are measured in dollars per week. Vanguard’s marketing director is comfortable using parameter estimates that are statistically significant at 10 per cent level or better. . What sign does the marketing director expect a, b, and c to have? The marketing director expects a, b, c to have appositive or a negative sign. b. Interpret the coefficients a, b, c. Coefficient a = 175086. 0 when b and c are zero. b- is greater than zero, therefore, we reject the null hypothesis (Ho: b = 0) and accept the alternative hypothesis (Ha : b ; 0). At the 10 percent significance level, the t-statistic (2. 63) is greater than the critical t-value (1. 697). c- co-efficient is negative and closer to and less than zero; the standard error is greater than the c-coefficient of the parameter estimate, R.
The t-statistic/t-ratio (-1. 73)) is smaller than the critical value (1. 697) at 33 degrees of freedom. c. Does Vanguard’s advertising expenditure have a statistically significant effect on sales of Bright Side detergent? Explain, using the appropriate p-value. Yes, Vanguard’s advertising expenditure has a statistically significant effect on sales because: Given the t-ratio, 2. 74, for advertising (b), the lowest level of significance for the co-efficient, b, is 1. 28 % and it’s less than the 10% significance level. This means there is 1. 28 chance that advertising expenditure does not affect sales of Bright Side laundry detergent. . Advertising by Vanguard’s three largest rivals is not statistically significant when using the t-statistic to test for significance. But, using the appropriate p-value , we learn that the p-value(0. 0927) from the regression output is less than the level of significance at 10 %, therefore we reject the null hypothesis (H0 : c=0), and we can conclude that advertising for the three biggest rival detergents?? is somewhat statistically significant. e. About 77. 53 % (1-R2) of the total variation in the sales of Bright Side remains unexplained.
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To increase the explanatory power of the sales equation the marketing director can add more explanatory variables and even replace the advertising expenditures of its biggest three rivals(R) or just increase the sample size. We can add consumer income(I), and price(P) as explanatory variable in this regression equation. f. What is the expected level of sales of each week when Vanguard spends $40000 p/w and the combined advertising expenditures for the three rivals are $100,000 per week. S = a + bA + cR S = 175086. 0 +0. 8550($40,000) + (-0. 284)*($100,000) S = 175086. 0 + 34200 ??? (28400) S = $180886