A motion picture industry analyst is studying movies based on epic novels. The f

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.

(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.)

%

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

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.)

+  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.)

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

Explanation / Answer

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

2        3.28      0.167            

3        4.60    0.004

4        1.54    0.175

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

          lower limit      upper limit

2          1.49   5.83    

3        4.40       10.84           

4        -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

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%

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