Question

The owner of Showtime Movie Theaters, Inc., would like to predict weekly gross revenue as a function of advertising expenditures. Historical data for a sample of eight weeks follow.

Weekly Gross Revenue ($1,000s)	Television Advertising ($1,000s)	Newspaper Advertising ($1,000s)
96	5.0	1.5
90	2.0	2.0
95	4.0	1.5
92	2.5	2.5
95	3.0	3.3
94	3.5	2.3
94	2.5	4.2
94	3.0	2.5

(a)

Develop an estimated regression equation with the amount of television advertising as the independent variable. (Round your numerical values to two decimal places. Let x₁ represent the amount of television advertising in $1,000s and y represent the weekly gross revenue in $1,000s.)

ŷ =

(b)

Develop an estimated regression equation with both television advertising and newspaper advertising as the independent variables. (Round your numerical values to two decimal places. Let x₁ represent the amount of television advertising in $1,000s, x₂ represent the amount of newspaper advertising in $1,000s, and y represent the weekly gross revenue in $1,000s.)

ŷ =

Answer 1

a.

Sum of X = 25.5
Sum of Y = 750
Mean X = 3.1875
Mean Y = 93.75
Sum of squares (SSX) = 6.4688
Sum of products (SP) = 10.375

Regression Equation = ŷ = bX + a

b = SP/SSX = 10.38/6.47 = 1.60

a = MY - bMX = 93.75 - (1.6*3.19) = 88.64

ŷ = 1.60X + 88.64

b.

Sum of X1 = 25.5
Sum of X2 = 19.8
Sum of Y = 750
Mean X1 = 3.1875
Mean X2 = 2.475
Mean Y = 93.75
Sum of squares (SSX1) = 6.4688
Sum of squares (SSX2) = 5.815
Sum of products (SPX1Y) = 10.375
Sum of products (SPX2Y) = -0.25
Sum of products (SPX1X2) = -3.4125

Regression Equation = ŷ = b1X1 + b2X2 + a

b1 = ((SPX1Y)*(SSX2)-(SPX1X2)*(SPX2Y)) / ((SSX1)*(SSX2)-(SPX1X2)*(SPX1X2)) = 59.48/25.97 = 2.29

b2 = ((SPX2Y)*(SSX1)-(SPX1X2)*(SPX1Y)) / ((SSX1)*(SSX2)-(SPX1X2)*(SPX1X2)) = 33.79/25.97 = 1.30

a = MY - b1MX1 - b2MX2 = 93.75 - (2.29*3.19) - (1.3*2.48) = 83.23

ŷ = 2.29X1 + 1.30X2 + 83.23