Question

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 3800 50 Hisense 2800 45 Hitachi 2700 35 JVC 3000 40 LG 3500 45 Maxent 2000 28 Panasonic 4000 57 Phillips 3200 48 Proview 2000 22 Samsung 3000 30 Use the above data to develop and estimated regression equation. 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 $3600. Perform test of significance for slope coefficient.

Answer 1

a).necessary table for finding regression equation:-

price(x)	score(y)	x^2	y^2	xy
3800	50	14440000	2500	190000
2800	45	7840000	2025	126000
2700	35	7290000	1225	94500
3000	40	9000000	1600	120000
3500	45	12250000	2025	157500
2000	28	4000000	784	56000
4000	57	16000000	3249	228000
3200	48	10240000	2304	153600
2000	22	4000000	484	44000
3000	30	9000000	900	90000
sum=30000	sum=400	sum=94060000	sum=17096	sum=1259600

so, the regression equation be:-

b). necessary table for calculation of coefficient of determination:-

score(y)	predicted value ()
50	51.74384	137.9178	100
45	37.06404	8.619865	25
35	35.59606	19.3947	25
40	40	5.05E-29	0
45	47.3399	53.87415	25
28	25.3202	215.4966	144
57	54.6798	215.4966	289
48	42.93596	8.619865	64
22	25.3202	215.4966	324
30	40	5.05E-29	100
sum=400		sum=874.916	sum=1096

coefficient of determination:-

correlation coefficient:-

relation between correlation coefficient and coefficient of determination:-

hence we can understand that,

(correlation coefficient)^2 = coefficient of determination.

c).the explanatory power of the model :-

79.83 % of the score can be explained by the independent variable, i.e, price.

d).the overall score for a 42-inch plasma television with a price of $3600 be:-

= -4.04 + (0.01468 * 3600 ) = $ 48.808

e).test of significance for slope coefficient:-

score(y)	predicted value ()		residual=()	residual^2
50	51.74384	640000	-1.74384	3.040986
45	37.06404	40000	7.935961	62.97947
35	35.59606	90000	-0.59606	0.355286
40	40	0	-7.1E-15	5.05E-29
45	47.3399	250000	-2.3399	5.475139
28	25.3202	1000000	2.679803	7.181344
57	54.6798	1000000	2.320197	5.383314
48	42.93596	40000	5.064039	25.6445
22	25.3202	1000000	-3.3202	11.02371
30	40	0	-10	100
sum=400		sum=4060000		sum=221.084

hypothesis:-

test statistic be:-

df = (n-2) = (10-2) = 8

p value = 0.0004 ( from p value calculator , df= 8 , t score = 5.625 , both sided test)

decision:-

p value = 0.0004 <0.05 ( I AM CONSIDERING LEVEL OF SIGNIFICANCE = 0.05)

so, we have enough evidence to reject the null hypothesis.

hence we conclude that the slope is significant (different from 0).

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