The following data show the brand, price , and the overall score for 6 stereo he

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 y=23.172+0.318x, where x=price and y=overall score.

Round your answers to three decimal places.

a.

b. Compute the coefficient of determination r^2.

Comment on the goodness of fit.

The input in the box below will not be graded, but may be reviewed and considered by your instructor.

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

Brand Price Score Bose 180 75 Scullcandy 150 72 Koss 85 62 Phillips/O'Neill 70 57 Denon 70 40 JVC 45 24

Explanation / Answer

Line of Regression Y on X i.e Y = bo + b1 X

calculation procedure for regression

mean of X = X / n = 100

mean of Y = Y / n = 55

(Xi - Mean)^2 = 13950

(Yi - Mean)^2 = 1928

(Xi-Mean)*(Yi-Mean) = 4440

b1 = (Xi-Mean)*(Yi-Mean) / (Xi - Mean)^2

= 4440 / 13950

= 0.31828

bo = Y / n - b1 * X / n

bo = 55 - 0.31828*100 = 23.17204

value of regression equation is, Y = bo + b1 X

Y'=23.172+0.318* X

PART A.

SST = 1,928.000, SSR = 1,413.161 , SSE = 514.839

c.

calculation procedure for correlation

sum of (x) = x = 600

sum of (y) = y = 330

sum of (x^2)= x^2 = 73950

sum of (y^2)= y^2 = 20078

sum of (x*y)= x*y = 37440

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

= 37440 - [ 6 * (600/6) * (330/6) ]/6- 1

= 740

and now to calculate r( x,y) = 740/ (SQRT(1/6*37440-(1/6*600)^2) ) * ( SQRT(1/6*37440-(1/6*330)^2)

=740 / (48.218*17.926)

=0.856

value of correlation is =0.856

coeffcient of determination = r^2 = 0.733

properties of correlation

1. If r = 1 Corrlation is called Perfect Positive Corrlelation

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 = 0.8561> 0 ,perfect positive correlation

X Y (Xi - Mean)^2 (Yi - Mean)^2 (Xi-Mean)*(Yi-Mean) 180 75 6400 400 1600 150 72 2500 289 850 85 62 225 49 -105 70 57 900 4 -60 70 40 900 225 450 45 24 3025 961 1705

Related Questions

Navigate

• Browse (All)
• Browse T
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.