Question

Many movies are released each year and it would be interesting to be able to predict the Total Gross Revenues (in $1,000,000) from the box office based on a few predictors. The following predictors have been identified for 70 movies:

• BUDGET: Estimated budget in $1,000,000
• LENGTH: The length of each movie in minutes
• SCREENS: Number of Screens on Opening Weekend
• AWARDS: Number of Award nominations of entire cast in their careers
• GENRE: Type of movie: Action, Comedy or Drama recoded as Genre 1 and Genre 2
  • Genre1 = 1 if Action, 0 if not
  • Genre2 = 1 if Comedy, 0 if not

                       

            The following partial output has been obtained:

	Coefficients	Standard Error	t Stat
Intercept	-4.039	30.735
Budget	0.803	0.154
Length	-0.433	0.242
Screens	0.013	0.005
Awards	1.390	1.049
Genre 1	4.777	2.032
Genre 2	2.732	13.646

ANOVA
	df	SS	MS	F
Regression			39.5
Residual		170.2
Total

Based on the above partial printout, answer the following questions:

1. What is the regression model?
2. Interpret the following coefficients: -0.433 and 4.777.
3. Is there sufficient evidence at the 5% level of significance to conclude that the model is useful at predicting Total Gross Revenues?
4. Determine the adjusted coefficient of determination and explain its meaning in the context of the problem.
5. What is the standard error of the estimate?
6. Does the “genre” of movie have a significant impact (at 5%) on the Total Gross Revenues? Justify.
7. Estimate the Total Gross Revenues for a Drama movie of 90 minutes produced with a $25,000,000 budget with a cast of actors who were nominated 6 times for awards. In addition, that movie was shown on 2,000 screens in the first weekend.

Answer 1

First we fill the missing values in the table as:

t-Stat=Coefficients/ standard error of coefficients
i.e. t-Stat=betahat/SE(betahat)
where, SE=Standard Error

ANOVA table:
GIVEN: MSregression=39.5, SSresidual=170.2
n=70 (total number of movies)
k=6 (total number of predictors)
p=k+1=7 (total number of parameters)
df(Total)=n-1=69
df(Regression)=k=6
df(Residual)=n-p=70-7=63
MSregression = SSRegression/df(Regression)
Therefore, SSregression=MSRegression*df(Regression)=39.5*6=237

MSResidual= SSResidual/df(Residual)=170.2/63=2.701587
Total SS= SSregression + SSresidual = 237 + 170.2 = 407.2
F=MSregression / MSresidual = 39.5 / 2.701587 =14.62103

Completed table is as follows:

	beta_hat	SE(betahat)		bhat/SE(bhat)
	Coefficients	Standard Error	ratio	t Stat	calc t	p-value	Formula in Excel
Intercept	-4.039	30.735	-4.039/30.735	-0.13141	2.296237	0.895867	TDIST(ABS(D3),63,2)
Budget	0.803	0.154	0.803/0.154	5.214286	2.296237	2.18E-06	TDIST(ABS(D4),63,2)
Length	-0.433	0.242	-0.433/0.242	-1.78926	2.296237	0.07838	TDIST(ABS(D5),63,2)
Screens	0.013	0.005	0.013/0.005	2.6	2.296237	0.0116	TDIST(ABS(D6),63,2)
Awards	1.39	1.049	1.39/01.049	1.325071	2.296237	0.189933	TDIST(ABS(D7),63,2)
Genre 1	4.777	2.032	4.777/2.032	2.350886	2.296237	0.021872	TDIST(ABS(D8),63,2)
Genre 2	2.732	13.646	2.732/13.646	0.200205	2.296237	0.841965	TDIST(ABS(D9),63,2)
TINV(0.025,63)=2.296237			FINV(0.05,6,63)=2.246408

ANOVA
	df	SS	MS	F
Regression	6	237	39.5	14.62103
Residual	63	170.2	2.701587
Total	69	407.2

Based on the above partial printout, answer the following questions:

a) What is the regression model?
Regression equation is given as:
## y=TGR=Total Gross Revenue
y= Intercept+ beta1*Budget+ beta2*Length+ beta3*Screens+ beta4*Awards+ beta5*Genre 1 + beta6*Genre 2
y= -4.039+ 0.803*BUDGET -0.433*LENGTH+ 0.013*SCREENS+ 1.390*AWARDS+ 4.777*GENRE1+ 2.732*GENRE2

b) Interpret the following coefficients: -0.433 and 4.777.
X=Independent variable/ Regressor
Y=Dependent variable / Response

• Positive sign means that if X increases, Y also increases and if X decreases Y also decreases. [Movement in Same direction]
• Negative sign of the coefficients means that if X increases Y decreases and if X decreases, Y increases.[Movement in OPPOSITE direction]

Here, (-0.433) indicates that if Length decreases , the revenue (response) increases and vice-versa.
Here, 4.777 indicates that: If Genre 1=1, then intercept will be (-4.039+4.777=0.738) and if Genre 1=0, then intercept remains unchanged.

c) Is there sufficient evidence at the 5% level of significance to conclude that the model is useful at predicting Total Gross Revenues?
H0: All beta_i's equal to zero i.e. regressors do not significantly contribute to the model.
Reject H0 if F > F(alpha,k,n-p)
Here, F(alpha,k,n-p)=FINV(0.05,6,63)=2.246408. Calculated F=14.62103
Calculated F > tabulated F, Thus, we Reject H0.
Yes, there sufficient evidence at the 5% level of significance to conclude that the model is useful at predicting Total Gross Revenues.

d) Determine the adjusted coefficient of determination and explain its meaning in the context of the problem.
## adjusted coefficient of determination = R2Adj

R-squared measures the proportion of the variation in your dependent variable (Y) explained by independent variables (X) for a linear regression model.The adjusted R-squared adjusts the statistic for the number of independent variables in the model. Importantly, its value increases only when the new term improves the model fit more than expected by chance alone.
R2Adj= 1 - [( SSresidual/ (n-p) ) / ( TotalSS / (n-1) )]= 1 - [ (170.2/63) / (407.2/69) ]=0.542216

54.2216 % of variation in the model is explianed by the regressors.

e) What is the standard error of the estimate?
Sigmasquare hat=MSresidual=2.701587
standard error of the estimate=SE=sigmahat=squareroot(2.701587) = 1.643651

f) Does the “genre” of movie have a significant impact (at 5%) on the Total Gross Revenues? Justify.
For Genre 1, calculated t-Stat > Tabulated t, we Reject H0 and conclude that Genre 1 is significant. [Also, p-value=0.021872 which is less than alpha=0.05, thus we Reject H0.]
BUT, for Genre 2, calculated t-Stat < Tabulated t, we Accept H0 and conclude that Genre 2 is insignificant. [Also, p-value=0.841965 which is greater than alpha=0.05, thus we Accept H0.]

We can conclude that Genre is significant.

g) Estimate the Total Gross Revenues for a Drama movie of 90 minutes produced with a $25,000,000 budget with a cast of actors who were nominated 6 times for awards. In addition, that movie was shown on 2,000 screens in the first weekend.

Since, it is drama movie, Genre1=Genre2=0.

Put Budget=$ 25 (in $1,000,000), Length=90, Awards=6, Screens=2000, Genre 1=0, Genre 2=0 in REGRESSION EQUATION:
y= -4.039+ 0.803*(25) -0.433*(90)+ 0.013*(2000)+ 1.390*(6)+ 4.777*(0)+ 2.732*(0)
y=11.406

Estimated Total Gross Revenue is 11.406 (in $1,000,000) i.e. $11,406,000.