Respuesta :
Answer:
Step-by-step explanation:
Hello!
You have the Data for three variables of interest:
"Weekly Gross Revenue"
"Television Advertising"
"Newspaper Advertising"
The owners of the Movie Theaters want to estimate the weekly gross revenue as a function of advertising expenditures.
For all asked regression the dependent variable will be Y: Weekly Gross Revenue
a)Using X: the amount of Television Advertising, as the independent variable, you have to test the simple linear regression.
The first step is to estimate the regression model
E(Yi)= α + βXi
Then ^Yi= a + bXi
Where "a" is the estimate of the intercept and "b" is the estimate of the slope.
Using a statistic software I've calculated the simple linear regression as:
a= -45.43
b= 40.06
^Yi= -45.43 + 40.06Xi
To test if there is a significant relationship between television advertising and weekly gross revenue you have to test the population slope of the regression, the hypotheses are:
H₀: β = 0
H₁: β ≠ 0
α: 0.05
You have two ways to test if the regression is significant, you either use a two-tailed t-test or a one-tailed F-test. They are two different distributions and test but you either one you can reach the same result.
I'll use the t-test for this item and the F-test for the later tests.
[tex]t= \frac{b-\beta }{Sb} ~ t_{n-2}[/tex]
[tex]t_{H_0}= \frac{40.06-0}{14.64}= 2.74[/tex]
The p-value for this test is 0.0339
Using the p-value approach the decision rule is:
If p-value ≤ α, the decision is to reject the null hypothesis.
If the p-value > α, the decision is to not reject the null hypothesis.
The p-value: 0.0339 is less than α: 0.05, the decision is to reject the null hypothesis.
The conclusion is that there is a significant relationship between television advertising and weekly gross revenue.
Looking at the estimated value of the slope, these two variables may have a direct relationship, i.e. every time the amount of television advertising is increased, the weekly gross revenue increases too. (This is only a supposition, without a propper hypothesis test you cannot conclude anything)
b) To know what % of the variation of the weekly gross revenue is explained by the model you have to calculate the coefficient of determination.
For item a) the coefficient is:
R²= 0.56
This means that 56% of the variability of the weekly gross revenue is explained by the amount of television advertisement under the estimated model: ^Yi= -45.43 + 40.06Xi
c) This time you have to develop a regression equation using two independent variables, be:
X₁: Amount of Television Advertising
X₂: Amount of Newspaper Advertising
The multiple regression model will be
E(Yi)= α + β₁X₁ + β₂X₂
α is the intercept ⇒ its estimator will be a
β₁ is the slope corresponding to X₁ ⇒ its estimator will be b₁
β₂ is the slope corresponding to X₂ ⇒ its estimator will be b₂
The estimated multiple regression model is ^Y= -42.57 + 22.40X₁ + 19.50X₂
The hypotheses for the overall regression are:
H₀: β₁ = β₂ = 0
H₁: At least one βi ≠ 0 ∀ i= 1, 2
α: 0.05
For this hypothesis test is best to use the F-test
[tex]F= \frac{MSreg}{MSerror}~~F_{DFreg;DFerror}[/tex]
[tex]F_{H_0}= \frac{14215.57}{413.37}= 34.39[/tex]
p-value: 0.0012
The p-value is less than α, the decision is to reject the null hypothesis.
Using a significance level of 5%, the overall regression is statistically significant.
For the single hypotheses I'll use the t-student and p-value approach:
1) Intercept
H₀: α = 0
H₁: α ≠ 0
α: 0.05
[tex]t_{H_0}= -1.49[/tex]
p-value: 0.1961
The p-value is greater than the level of significance, the decision is to not reject the null hypothesis.
2) Slope for X₁
H₀: β₁ = 0
H₁: β₁ ≠ 0
α: 0.05
[tex]t_{H_0}= 3.16[/tex]
p-value: 0.0252
The p-value is less than the significance level, the decision is to reject the null hypothesis.
Using a significance level of 5%, the regression is significant i.e. the amount of television advertising modifies the average weekly gross revenue.
3) Slope for X₂
H₀: β₂ = 0
H₁: β₂ ≠ 0
α: 0.05
[tex]t_{H_0}= 5.27[/tex]
p-value: 0.0033
The p-value is less than the significance level, the decision is to reject the null hypothesis.
Using a significance level of 5%, the regression is significant, i.e. the amount of newspaper advertising modifies the average weekly gross revenue.
d) The corresponding coefficient of determination for the multiple regression is R²= 0.93
93% of the variability of the average weekly gross revenue is explained jointly by the amount of television advertisement and newspaper advertisement under the estimated model: ^Y= -42.57 + 22.40X₁ + 19.50X₂
e) and f)
Comparing the result in a) and c) you can say that both independent variables are good to explain the dependent variable.
Comparing both R², we can say that the amount of television advertisement alone isn't a good explanatory variable for the variability weekly gross revenue but together with the amount of newspaper advertisement it becomes a good explanatory variable.
I hope this helps!
