Question

Consumer Reportsprovided extensive testing and ratings for more than 100 HDTVs. An overall score, based primarily on picture quality, was developed for each model. In general, a higher overall score indicates better performance. The following (hypothetical) data show the price and overall score for the ten 42-inch plasma televisions (Consumer Reportdata slightly changed here):

Brand	Price (X)	Score (Y)
Dell	2900	50
Hisense	2800	52
Hitachi	2700	45
JVC	3500	60
LG	3300	56
Maxent	2000	30
Panasonic	4200	68
Phillips	3100	56
Proview	2500	35
Samsung	3000	48

Use the above data to develop and estimated regression equation and interpret the coefficients. Compute Coefficient of Determination and correlation coefficient and show their relation. Interpret the explanatory power of the model. Estimate the overall score for a 42-inch plasma television with a price of $3400. Finally, test the significance of the slope coefficient. (Note that you need to answer all parts of the question and provide necessary interpretations to get full points).

Question is from Eco 302.

Answer 1

Solution :
Xmean = (2900 + 2800 + 2700 + 3500 + 3300 + 2000 + 4200 + 3100 + 2500 + 3000)/10
= 30000/10
Xmean = 3000

Ymean = (50 + 52 + 45 + 60 + 56 + 30 + 68 + 56 + 35 + 48 )/10
= 500/10
Ymean = 50

Slope =(∑y)(∑x²) - (∑x)(∑xy)n(∑x²) - (∑x)²
∑y = 50 + 52 + 45 + 60 + 56 + 30 + 68 + 56 + 35 + 48
∑y = 500
∑x² = (2900)² + ( 2800)² + ( 2700)² + ( 3500)² + ( 3300)² + ( 2000)² + ( 4200)² + ( 3100)² + ( 2500)² + ( 3000)²
= 8410000 + 7840000 + 7290000 + 12250000 + 10890000 + 4000000 + 17640000 + 9610000 + 6250000 + 9000000
∑x² = 93180000
∑x = 2900 + 2800 + 2700 + 3500 + 3300 + 2000 + 4200 + 3100 + 2500 + 3000
∑x = 30000
∑xy = (2900 x 50) + ( 2800 x 52) + ( 2700 x 45) + ( 3500 x 60) + ( 3300 x 56) + ( 2000 x 30) + ( 4200 x 68) + ( 3100 x 56) + ( 2500 x 35) + ( 3000 x 48 )
∑xy = 145000 + 145600 + 121500 + 210000 + 184800 + 60000 + 285600 + 173600 + 87500 + 144000
∑xy = 1557600
Apply the values in above formula
Slope =(500 x 93180000) - (30000 x 1557600))(10 x 93180000) - (30000)²
=46590000000 - 46728000000931800000 - 900000000
=-13800000031800000
Slope = -4.3396

Intercept =n(∑xy) - (∑x)(∑y)n(∑x²) - (∑x)²
=10(1557600) - (30000 x 500)(10 x 93180000) - (30000)²
=15576000 - 15000000931800000 - 900000000
=57600031800000
Intercept = 0.0181

Regression equation = Intercept + Slope x
Regression equation = -4.3396 + 0.0181 x

Estimate the overall score for a 42-inch plasma television with a price of $3400.

Putting the value in the regression equation we obtain. 57.20.

SUMMARY OUTPUT
Regression Statistics
Multiple R	0.9508
R Square	0.9041
Adjusted R Square	0.8921
Standard Error	3.7195
Observations	10

Since the R Squared value is 0.90 hence there is a very strong correlation between price and score.