Question

The following data show the brand, price ($), and the overall score for 6 stereo headphones that were tested by Consumer Reports. The overall score is based on sound quality and effectiveness of ambient noise reduction. Scores range from 0 (lowest) to 100 (highest). The estimated regression equation for these data is  = 20.397 + 0.346 x, where x = price ($) and y = overall score.

Brand	Price ($)	Score
Bose	170	75
Scullcandy	150	72
Koss	95	62
Phillips/O'Neill	70	56
Denon	60	40
JVC	55	25

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 =

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

r =

Answer 1

Solution: First, we will prepare the following table by following the below-mentioned steps:

Step 1: Find the Mean Values for Xi and Yi

Mean (Xi) = Sum of Xi / n = 600 / 6 = 100 (Nore: n = Sample size = 6)

Mean (Yi) = Sum of Yi / n = 330 / 6 = 55

Step 2: Find Column 3 = Yi - Mean (Yi) (i.e., Subtract the Yi values from Mean (Yi)

Step 3: Find column 4 = (Yi - Mean (Yi)) ^ 2 = (Col 3) ^ 2

Step 4: Find column 5 = E(Y) = 20.397 + 0.346 Xi

Step 5: Find column 6 = Yi - E(Y)

Step 6: Find column 7 = [Yi - E(Y)] ^ 2

Now, we will answer the given quesitions one by one:

Answer a)

Where

SST = Sum of {(Yi - Mean (Yi)) ^ 2}, = Sum of Column 4

SSE = Sum of { [Yi - E(Y)] ^ 2} = Sum of Column 7, and

SSR = SST - SSE

Answer b)

Coefficient of Determination = r^2 = SSR / SST = 1430.8137 / 1864 = 0.7676

Comment on the goodness of fit = Here the value of the coefficient of determination is near to 1, which refers to this model fairly estimates and explains the variations in the value of score (y) due to change in the price (x)

Answer c)

Correlation Coefficient = SQRT ( r^) = SQRT (0.7676) = 0.8716