Question

a. An experiment was performed on a certain metal to determine if the strength is a function of heating time (hours). Results based on 25 metal sheets are given below. Use the simple linear regression model.
∑X = 50
∑X² = 200
∑Y = 75
∑Y² = 1600
∑XY = 400

Find the estimated y intercept and slope. Write the equation of the least squares regression line and explain the coefficients. Estimate Y when X is equal to 4 hours. Also determine the standard error, the Mean Square Error, the coefficient of determination and the coefficient of correlation. Check the relation between correlation coefficient and Coefficient of Determination. Test the significance of the slope.

b. 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 and perform significance test for the slope.

Answer 1

a)

Ʃx =	50
Ʃy =	75
Ʃxy =	400
Ʃx² =	200
Ʃy² =	1600
Sample size, n =	25
x̅ = Ʃx/n = 50/25 =	2
y̅ = Ʃy/n = 75/25 =	3
SSxx = Ʃx² - (Ʃx)²/n = 200 - (50)²/25 =	100
SSyy = Ʃy² - (Ʃy)²/n = 1600 - (75)²/25 =	1375
SSxy = Ʃxy - (Ʃx)(Ʃy)/n = 400 - (50)(75)/25 =	250

Slope, b = SSxy/SSxx = 250/100 = 2.5

y-intercept, a = y̅ -b* x̅ = 3 - (2.5)*2 = -2

Regression equation :

ŷ = -2 + (2.5) x

Predicted value of y at x = 4

ŷ = -2 + (2.5) * 4 = 8

Sum of Square error, SSE = SSyy -SSxy²/SSxx = 1375 - (250)²/100 = 750

Standard error, se = √(SSE/(n-2)) = √(750/(25-2)) = 5.71040

MSE = (SSE/(n-2)) = (750/(25-2)) = 32.6087

Correlation coefficient, r = SSxy/√(SSxx*SSyy) = 250/√(100*1375) = 0.6742

Coefficient of determination, r² = (SSxy)²/(SSxx*SSyy) = (250)²/(100*1375) = 0.4545

Correlation coefficient is the square root of Coefficient of Determination.

Slope Hypothesis test:

Null and alternative hypothesis:

Ho: β₁ = 0 ; Ha: β₁ ≠ 0

α = 0.05

Slope, b = 2.5

Test statistic:

t = b/(se/√SSxx) = 4.3780

df = n-2 = 23

p-value = T.DIST.2T(ABS(4.378), 23) = 0.0002

Conclusion:

p-value < α Reject the null hypothesis.

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b)

Ʃx =	30000
Ʃy =	400
Ʃxy =	1259600
Ʃx² =	94060000
Ʃy² =	17096
Sample size, n =	10
x̅ = Ʃx/n = 30000/10 =	3000
y̅ = Ʃy/n = 400/10 =	40
SSxx = Ʃx² - (Ʃx)²/n = 94060000 - (30000)²/10 =	4060000
SSyy = Ʃy² - (Ʃy)²/n = 17096 - (400)²/10 =	1096
SSxy = Ʃxy - (Ʃx)(Ʃy)/n = 1259600 - (30000)(400)/10 =	59600

Slope, b = SSxy/SSxx = 59600/4060000 = 0.0146798

y-intercept, a = y̅ -b* x̅ = 40 - (0.01468)*3000 = -4.0394089

Regression equation :

ŷ = -4.0394 + (0.0147) x

Predicted value of y at x = 3600

ŷ = -4.0394 + (0.0147) * 3600 = 48.8079

Correlation coefficient, r = SSxy/√(SSxx*SSyy) = 59600/√(4060000*1096) = 0.8935

Coefficient of determination, r² = (SSxy)²/(SSxx*SSyy) = (59600)²/(4060000*1096) = 0.7983

Sum of Square error, SSE = SSyy -SSxy²/SSxx = 1096 - (59600)²/4060000 = 221.083744

Standard error, se = √(SSE/(n-2)) = √(221.08374/(10-2)) = 5.25694

Slope Hypothesis test:

Null and alternative hypothesis:

Ho: β₁ = 0 ; Ha: β₁ ≠ 0

α = 0.05

Test statistic:

t = b/(se/√SSxx) = 5.6266

df = n-2 = 8

p-value = T.DIST.2T(ABS(5.6266), 8) = 0.0005

Conclusion:

p-value < α Reject the null hypothesis.