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	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.

Answer 1

x	y	(x-x̅)²	(y-ȳ)²	(x-x̅)(y-ȳ)
2900	50	10000.00	0.00	0.00
2800	52	40000.00	4.00	-400.00
2700	45	90000.00	25.00	1500.00
3500	60	250000.00	100.00	5000.00
3300	56	90000.00	36.00	1800.00
2000	30	1000000.00	400.00	20000.00
4200	68	1440000.00	324.00	21600.00
3100	56	10000.00	36.00	600.00
2500	35	250000.00	225.00	7500.00
3000	48	0.00	4.00	0.00

	ΣX	ΣY	Σ(x-x̅)²	Σ(y-ȳ)²	Σ(x-x̅)(y-ȳ)
total sum	30000	500	3180000	1154.0	57600.00
mean	3000.00	50.00	SSxx	SSyy	SSxy

sample size ,   n =   10          
here, x̅ = Σx / n=   3000.00   ,     ȳ = Σy/n =   50.00  
                  
SSxx =    Σ(x-x̅)² =    3180000.0000          
SSxy=   Σ(x-x̅)(y-ȳ) =   57600.0          
                  
estimated slope , ß1 = SSxy/SSxx =   57600.0   /   3180000.000   =   0.0181
                  
intercept,   ß0 = y̅-ß1* x̄ =   -4.3396          
                  
so, regression line is   Ŷ =   -4.3396   +   0.0181   *x

................

intercept : -4.3396

if x= 0, then the score (Y) will be - 4.3396

slope = 0.0181

if x is increase by 1 unit, score (Y) will be increased by 0.0181

.........................

correlation coefficient ,    r = Sxy/√(Sx.Sy) =   0.9508
      
cofficient of determination ,R² =    (Sxy)²/(Sx.Sy) =    0.9041
cofficient of determination = (correlation cofficient )2

....................

Predicted Y at X=   3400   is                  
Ŷ =   -4.33962   +   0.018113   *   3400   =   57.245
..............

slope hypothesis test               tail=   2
Ho:   ß1=   0          
H1:   ß1╪   0          
n=   10              
alpha =   0.05              
estimated std error of slope =Se(ß1) = Se/√Sxx =    3.720   /√   3180000.00   =   0.0021
                  
t stat = estimated slope/std error =ß1 /Se(ß1) =    0.0181   /   0.0021   =   8.6840
                  
t-critical value=    2.306   [excel function: =T.INV.2T(α,df) ]          
Degree of freedom ,df = n-2=   8              
p-value =    0.0000              
decison :    p-value<α , reject Ho              
Conclusion:   Reject Ho and conclude that slope is significantly different from zero              

..............

Please revert back in case of any doubt.

Please upvote. Thanks in advance.