Problem 4-9 Dixie Showtime Movie Theaters, Inc., owns and operates a chain of ci

Question

Problem 4-9
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.
Click on the webfile logo to reference the data.

Market   Weekly Gross Revenue ($100s)   Television Advertising ($100s)   Newspaper Advertising ($100s)
Mobile   101.3    5.0 1.5
Shreveport   51.9      3.0      3.0
Jackson   74.8 4.0      1.5
Birmingham   126.2                                           4.3      4.3
Little Rock   137.8 3.6      4.0
Biloxi   101.4    3.5                                                      2.3
New Orleans   237.8      5.0      8.4
Baton Rouge   219.6    6.9 5.8
(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)
= + x
(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)
= + x1 + x2
   

Explanation / Answer

a.

slope (b) = (NXY - (X)(Y)) / (NX2 - (X)2)

= ((8*5059.28) - (35.3*1050.8)/((8*166.31 - (35.3)^2)

= (3381/84.39) = 40.06

a = (Y - b(X)) / N

(1050.8 - 40.06*35.3)/8 = - 45.43

Thus the regression equation is y = - 45.43 +40.06x

b. We calculate the "r squared" for this part. Using the function "RSQ" in excel, we get the r square as 0.5551. I means that 55.51% of the variation in the sample values of weekly gross revenue is explained by the model.

c. The equation here will be y = a+b1x1+b2x2

b1 = (x2^2)(x1y) - (x1x2)(x2y)/(x1^2)(x2^2) - (x1x2)^2

166.31

b1 = (157.48)(5059.28) - (145.46)(5018.13)/(166.31)(157.48) - (145.46)^2

= 66798.22/5031.887 = 13.27

b2 = (x1^2)(x2y) - (x1x2)(x1y)/(x1^2)(x2^2) - (x1x2)^2

= (166.31)(5018.13) - (145.46)(5059.28)/(166.31)(157.48) - (145.46)^2

= 98642.33/5031.887 = 19.60

a = mean of Y - b1 *mean of x1 - b2 * mean of x2

a = 131.35 - (13.27*4.41) - (19.6*3.85) = - 2.67

Thus the equation: y = - 2.76 +13.27x1+19.60x2

x y x*y x*x 5 101.3 506.5 25 3 51.9 155.7 9 4 74.8 299.2 16 4.3 126.2 542.66 18.49 3.6 137.8 496.08 12.96 3.5 101.4 354.9 12.25 5 237.8 1189 25 6.9 219.6 1515.24 47.61 Total 35.3 1050.8 5059.28 166.31

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