Question

Dixie Showtime Movie Theaters, Inc., owns and operates a chain of cinemas in several markets in the southern U.S. The owners would like to estimate weekly gross revenue as a function of advertising expenditures. Data for a sample of eight markets for a recent week follow.
Market Weekly Gross Revenue Television Advertising Newspaper Advertising
($100s) ($100s) ($100s)
Mobile 101.3 5 1.5
Shreveport 51.9 3 3
Jackson 74.8 4 1.5
Birmingham 126.2 4.3 4.3
Little Rock 137.8 3.6 4
Biloxi 101.4 3.5 2.3
New Orleans 237.8 5 8 .4
Baton Rouge 219.6 6.9 5.8

Required:
(a) Use the data to develop an estimated regression with the amount of television advertising as the independent variable. Let x represent the amount of television advertising. If required, round your answers to four decimal places. For subtractive or negative numbers use a minus sign even if there is a + sign before the blank. (Example: -300).
Test for a significant relationship between television advertising and weekly gross revenue at the 0.05 level of significance. What is the interpretation of this relationship?
(b) How much of the variation in the sample values of weekly gross revenue does the model in part (a) explain?
If required, round your answer to two decimal places.
(c) Use the data to develop an estimated regression equation with both television advertising and newspaper advertising as the independent variables.
Let x1 represent the amount of television advertising.
Let x2 represent the amount of newspaper advertising.
If required, round your answers to four decimal places. For subtractive or negative numbers use a minus sign even if there is a + sign before the blank. (Example: -300)
Is the overall regression statistically significant at the 0.05 level of significance? If so, then test whether each of the regression parameters β0, β1, and β2 is equal to zero at a 0.05 level of significance. What are the correct interpretations of the estimated regression parameters? Are these interpretations reasonable?
(d) How much of the variation in the sample values of weekly gross revenue does the model in part (c) explain?
If required, round your answer to two decimal places.
(e) Given the results in part (a) and part (c), what should your next step be? Explain.
(f) What are the managerial implications of these results?

Answer 1

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.

$t= \frac{b-\beta }{Sb} ~ t_{n-2}$

$t_{H_0}= \frac{40.06-0}{14.64}= 2.74$

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

$F= \frac{MSreg}{MSerror}~~F_{DFreg;DFerror}$

$F_{H_0}= \frac{14215.57}{413.37}= 34.39$

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

$t_{H_0}= -1.49$

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

$t_{H_0}= 3.16$

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

$t_{H_0}= 5.27$

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!