In: Statistics and Probability
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 all parts of the question and provide necessary interpretations.
ANSWER::
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
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