Question

10. Consumer Reports provided 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 Report data 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).

Answer 1

X	Y	XY	X²	Y²
2900	50	145000	8410000	2500
2800	52	145600	7840000	2704
2700	45	121500	7290000	2025
3500	60	210000	12250000	3600
3300	56	184800	10890000	3136
2000	30	60000	4000000	900
4200	68	285600	17640000	4624
3100	56	173600	9610000	3136
2500	35	87500	6250000	1225
3000	48	144000	9000000	2304
Ʃx =	Ʃy =	Ʃxy =	Ʃx² =	Ʃy² =
30000	500	1557600	93180000	26154

Sample size, n =	10
x̅ = Ʃx/n = 30000/10 =	3000
y̅ = Ʃy/n = 500/10 =	50
SSxx = Ʃx² - (Ʃx)²/n = 93180000 - (30000)²/10 =	3180000
SSyy = Ʃy² - (Ʃy)²/n = 26154 - (500)²/10 =	1154
SSxy = Ʃxy - (Ʃx)(Ʃy)/n = 1557600 - (30000)(500)/10 =	57600

Slope, b = SSxy/SSxx = 57600/3180000 = 0.0181132

y-intercept, a = y̅ -b* x̅ = 50 - (0.01811)*3000 = -4.339623

Regression equation :

ŷ = -4.3396 + (0.0181) x

Slope: With a unit increase price, the value of score increases by 0.0181 units

Y-intercept: It is value when x = 0. it is not reasonable in this case.

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Correlation coefficient, r = SSxy/√(SSxx*SSyy) = 57600/√(3180000*1154) = 0.9508

Coefficient of determination, r² = (SSxy)²/(SSxx*SSyy) = (57600)²/(3180000*1154) = 0.9041

90.41% variation in y is explained by the least squares model.

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Predicted value of y at x = 3400

ŷ = -4.3396 + (0.0181) * 3400 = 57.2453

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Null and alternative hypothesis:

Ho: β₁ = 0

Ha: β₁ ≠ 0

Sum of Square error, SSE = SSyy -SSxy²/SSxx = 1154 - (57600)²/3180000 = 110.6792453

Standard error, se = √(SSE/(n-2)) = √(110.67925/(10-2)) = 3.71953

Test statistic:

t = b/(se/√SSxx) = 8.6840

df = n-2 = 8

p-value = T.DIST.2T(ABS(8.684), 8) = 0.0000

Conclusion:

p-value < α Reject the null hypothesis. The slope is significant.