Question

The following data show the brand, price ($), and the overall score for 6 stereo headphones that were tested by Consumer Reports. The overall score is based on sound quality and effectiveness of ambient noise reduction. Scores range from 0 (lowest) to 100 (highest). The estimated regression equation for these data is  = 23.395 + 0.316 x, where x = price ($) and y = overall score.

Brand	Price ($)	Score
Bose	180	75
Scullcandy	150	72
Koss	95	61
Phillips/O'Neill	70	56
Denon	70	40
JVC	35	26

Round your answers to three decimal places.

a. Compute SST, SSR, and SSE.

SST
SSR
SSE

b. Compute the coefficient of determination r ². Comment on the goodness of fit.

r ² =

SelectThe least squares line provided a very good fitThe least squares line provided a bad fitItem 5

c. What is the value of the sample correlation coefficient?

r =

Answer 1

Line of Regression Y on X i.e Y = bo + b1 X
X	Y	(Xi - Mean)^2	(Yi - Mean)^2	(Xi-Mean)*(Yi-Mean)
180	75	6400	400	1600
150	72	2500	289	850
95	61	25	36	-30
70	56	900	1	-30
70	40	900	225	450
35	26	4225	841	1885

calculation procedure for regression
mean of X = ∑ X / n = 100
mean of Y = ∑ Y / n = 55
∑ (Xi - Mean)^2 = 14950
∑ (Yi - Mean)^2 = 1792
∑ (Xi-Mean)*(Yi-Mean) = 4725
b1 = ∑ (Xi-Mean)*(Yi-Mean) / ∑ (Xi - Mean)^2
= 4725 / 14950
= 0.316
bo = ∑ Y / n - b1 * ∑ X / n
bo = 55 - 0.316*100 = 23.395
value of regression equation is, Y = bo + b1 X
Y'=23.395+0.316* X          

a.

( X)	( Y)	X^2	Y^2	X*Y
180	75	32400	5625	13500
150	72	22500	5184	10800
95	61	9025	3721	5795
70	56	4900	3136	3920
70	40	4900	1600	2800
35	26	1225	676	910

calculation procedure for correlation
sum of (x) = ∑x = 600
sum of (y) = ∑y = 330
sum of (x^2)= ∑x^2 = 74950
sum of (y^2)= ∑y^2 = 19942
sum of (x*y)= ∑x*y = 37725
to caluclate value of r( x,y) = covariance ( x,y ) / sd (x) * sd (y)
covariance ( x,y ) = [ ∑x*y - N *(∑x/N) * (∑y/N) ]/n-1
= 37725 - [ 6 * (600/6) * (330/6) ]/6- 1
= 787.5
and now to calculate r( x,y) = 787.5/ (SQRT(1/6*37725-(1/6*600)^2) ) * ( SQRT(1/6*37725-(1/6*330)^2)
=787.5 / (49.917*17.282)
=0.913
value of correlation is =0.913
b.

coeffcient of determination = r^2 = 0.833
properties of correlation
1. If r = 1 Corrlation is called Perfect Positive Correlation
2. If r = -1 Correlation is called Perfect Negative Correlation
3. If r = 0 Correlation is called Zero Correlation
& with above we conclude that correlation ( r ) is = 0.9129> 0 ,perfect positive correlation              
c.

value of correlation is =0.913