Question

A motion picture industry analyst is studying movies based on epic novels. The following data were obtained for 10 Hollywood movies made in the past five years. Each movie was based on an epic novel. For these data, x₁ = first-year box office receipts of the movie, x₂ = total production costs of the movie, x₃ = total promotional costs of the movie, and x₄ = total book sales prior to movie release. All units are in millions of dollars.

x1	x2	x3	x4
85.1	8.5	5.1	4.7
106.3	12.9	5.8	8.8
50.2	5.2	2.1	15.1
130.6	10.7	8.4	12.2
54.8	3.1	2.9	10.6
30.3	3.5	1.2	3.5
79.4	9.2	3.7	9.7
91.0	9.0	7.6	5.9
135.4	15.1	7.7	20.8
89.3	10.2	4.5	7.9

(a) Generate summary statistics, including the mean and standard deviation of each variable. Compute the coefficient of variation for each variable. (Use 2 decimal places.)

	x	s	CV
x1			%
x2			%
x3			%
x4			% (b) For each pair of variables, generate the correlation coefficient r. Compute the corresponding coefficient of determination r2. (Use 3 decimal places.) r r2 x1, x2 x1, x3 x1, x4 x2, x3 x2, x4 x3, x4 What percent of the variation in box office receipts can be attributed to the corresponding variation in production costs? (Use 1 decimal place.) % (c) Perform a regression analysis with x1 as the response variable. Use x2, x3, and x4 as explanatory variables. Look at the coefficient of multiple determination. What percentage of the variation in x1 can be explained by the corresponding variations in x2, x3, and x4 taken together? (Use 1 decimal place.) % (d) Write out the regression equation. (Use 2 decimal places.) x1 = +  x2 +  x3 +  x4 If x2 (production costs) and x4 (book sales) were held fixed but x3 (promotional costs) were increased by 1.2 million dollars, what would you expect for the corresponding change in x1 (box office receipts)? (Use 2 decimal places.) (e) Test each coefficient in the regression equation to determine if it is zero or not zero. Use level of significance 5%. (Use 2 decimal places for t and 3 decimal places for the P-value.) t P-value β2 β3 β4 (f) Find a 90% confidence interval for each coefficient. (Use 2 decimal places.) lower limit upper limit β2 β3 β4 (g) Suppose a new movie (based on an epic novel) has just been released. Production costs were x2 = 11.4 million; promotion costs were x3 = 4.7 million; book sales were x4 = 8.1 million. Make a prediction for x1 = first-year box office receipts and find an 85% confidence interval for your prediction (if your software supports prediction intervals). (Use 1 decimal place.) prediction lower limit upper limit (h) Construct a new regression model with x3 as the response variable and x1, x2, and x4 as explanatory variables. (Use 2 decimal places.) x3 = +  x1 +  x2 +  x4 Suppose Hollywood is planning a new epic movie with projected box office sales x1 = 100 million and production costs x2 = 12 million. The book on which the movie is based had sales of x4 = 9.2 million. Forecast the dollar amount (in millions) that should be budgeted for promotion costs x3 and find an 80% confidence interval for your prediction. prediction lower limit upper limit

Answer 1

Result:

(a) Generate summary statistics, including the mean and standard deviation of each variable. Compute the coefficient of variation for each variable. (Use 2 decimal places.)

Descriptive statistics
	x1	x2	x3	x4
n	10	10	10	10
mean	85.24	8.74	4.90	9.92
sample standard deviation	33.79	3.89	2.48	5.17
coefficient of variation (CV)	39.64%	44.45%	50.62%	52.15%

(b) For each pair of variables, generate the correlation coefficient r. Compute the corresponding coefficient of determination r2. (Use 3 decimal places.)

	r	r2
x1,x2	0.917	0.841
x1,x3	0.930	0.865
x1,x4	0.475	0.226
x2,x3	0.790	0.624
x2,x4	0.429	0.184
x3,x4	0.299	0.089

What percent of the variation in box office receipts can be attributed to the corresponding variation in production costs? (Use 1 decimal place.)

84.1%

(c) Perform a regression analysis with x1 as the response variable. Use x2, x3, and x4 as explanatory variables. Look at the coefficient of multiple determination. What percentage of the variation in x1 can be explained by the corresponding variations in x2, x3, and x4 taken together? (Use 1 decimal place.)

96.7%

(d) Write out the regression equation. (Use 2 decimal places.)

x1 = 7.68        + 3.66*x2       + 7.62* x3 + 0.83* x4

If x2 (production costs) and x4 (book sales) were held fixed but x3 (promotional costs) were increased by 1.2 million dollars, what would you expect for the corresponding change in x1 (box office receipts)? (Use 2 decimal places.)

1.2*7.6211 =9.14

(e) Test each coefficient in the regression equation to determine if it is zero or not zero. Use level of significance 5%. (Use 2 decimal places for t and 3 decimal places for the P-value.)

          t           P-value

β2        3.28      0.167            

β3        4.60    0.004

β4        1.54    0.175

(f) Find a 90% confidence interval for each coefficient. (Use 2 decimal places.)

          lower limit      upper limit

β2          1.49   5.83    

β3        4.40       10.84           

β4        -0.22     1.88

(g) Suppose a new movie (based on an epic novel) has just been released. Production costs were x2 = 11.4 million; promotion costs were x3 = 4.7 million; book sales were x4 = 8.1 million. Make a prediction for x1 = first-year box office receipts and find an 85% confidence interval for your prediction (if your software supports prediction intervals). (Use 1 decimal place.)

prediction      

lower limit      77.6

upper limit   106.3     

Predicted values for: x1
				85% Confidence Interval		85% Prediction Interval
x2	x3	x4	Predicted	lower	upper	lower	upper
11.4	4.7	8.1	91.9478	84.7384	99.1573	77.5663	106.3294

(h) Construct a new regression model with x3 as the response variable and x1, x2, and x4 as explanatory variables. (Use 2 decimal places.)

x3 =    + x1    + x2    + x4

Suppose Hollywood is planning a new epic movie with projected box office sales x1 = 100 million and production costs x2 = 12 million. The book on which the movie is based had sales of x4 = 9.2 million. Forecast the dollar amount (in millions) that should be budgeted for promotion costs x3 and find an 80% confidence interval for your prediction.

prediction      

lower limit      4.21

upper limit      7.04

Regression Analysis
	R²	0.967
	Adjusted R²	0.950	n	10
	R	0.983	k	3
	Std. Error	7.541	Dep. Var.	x1
ANOVA table
Source	SS	df	MS	F	p-value
Regression	9,932.4635	3	3,310.8212	58.22	.0001
Residual	341.2005	6	56.8668
Total	10,273.6640	9
Regression output					confidence interval
variables	coefficients	std. error	t (df=6)	p-value	90% lower	90% upper
Intercept	7.6760	6.7602	1.135	.2995	-5.4603	20.8124
x2	3.6616	1.1178	3.276	.0169	1.4896	5.8336
x3	7.6211	1.6573	4.598	.0037	4.4006	10.8415
x4	0.8285	0.5394	1.536	.1754	-0.2196	1.8765

Regression Analysis
	R²	0.917
	Adjusted R²	0.876	n	10
	R	0.958	k	3
	Std. Error	0.873	Dep. Var.	x3
ANOVA table
Source	SS	df	MS	F	p-value
Regression	50.7838	3	16.9279	22.20	.0012
Residual	4.5762	6	0.7627
Total	55.3600	9
Regression output					confidence interval
variables	coefficients	std. error	t (df=6)	p-value	80% lower	80% upper
Intercept	-0.6499	0.8211	-0.792	.4588	-1.8321	0.5323
x1	0.1022	0.0222	4.598	.0037	0.0702	0.1342
x2	-0.2598	0.1883	-1.379	.2170	-0.5310	0.0114
x4	-0.0899	0.0639	-1.406	.2093	-0.1820	0.0022
Predicted values for: x3
				80% Confidence Interval		80% Prediction Interval
x1	x2	x4	Predicted	lower	upper	lower	upper
100	12	9.2	5.6264	4.9712	6.2816	4.2085	7.0442