Question

The following data show the brand, price ($), and the overall score for six stereo headphones that were tested by a certain magazine. 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

ŷ = 20.455 + 0.335x,

where x = price ($) and y = overall score.

Brand	Price ($)	Score
A	180	78
B	150	69
C	95	59
D	70	56
E	70	38
F	35	24

(1) Compute SST, SSR, and SSE. (Round your answers to three decimal places.)

SST=

SSR=

SSE=

(2a) Compute the coefficient of determination r². (Round your answer to three decimal places.)

r² = ?

(2b) Comment on the goodness of fit. (For purposes of this exercise, consider a proportion large if it is at least 0.55.)

(a) The least squares line provided a good fit as a large proportion of the variability in y has been explained by the least squares line.

(b) The least squares line did not provide a good fit as a large proportion of the variability in y has been explained by the least squares line.    

(c) The least squares line did not provide a good fit as a small proportion of the variability in y has been explained by the least squares line.

(d) The least squares line provided a good fit as a small proportion of the variability in y has been explained by the least squares line.

3. What is the value of the sample correlation coefficient? (Round your answer to three decimal places.)

Answer 1

The statistic software output for this problem is :

Simple linear regression results: Dependent Variable: Score Independent Variable: Price (5) Score = 20.454849+0.33545151 Price ($) Sample size: 6 R (correlation coefficient) = 0.92036632 R-q=0.84707417 Estimate of error standard deviation: 8.7136488 Parameter estimates: Parameter Estimate Std. Err. Intercept 20.454849 7.965071 Slope 0.33545151 0.07126552 Alternative DFT-Stat P-value 0 4 2.5680687 0.0621 +0 4 4.707066 0.0093 Analysis of variance table for regression model: Source DF SS MS F -stat P-value Model 1 1682.2893 1682.2893 22.15647 0.0093 Error 4 303.7107 75.927676 Total 5 1986

SST= 1986

SSR= 1682.289

SSE= 303.711

r ² = 0.847

(d) The least squares line provided a good fit as a small proportion of the variability in y has been explained by the least squares line.

sample correlation coefficient = 0.920