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The following data show the brand, price ($), and the overall score for six stereo headphones that were tested by a certain magazine. The overall score is based on sound quality and effectiveness of ambient noise reduction. Scores range from 0 (lowest) to 100 (highest). The estimated regression equation for these data is û = 22.793 + 0.322x, where x = price ($) and y = overall score. Brand Price ($) Score A 180 78 B 150 69 C 95 61 D 70 56 E 70 40 F 35 26 (a) Compute SST, SSR, and SSE. (Round your answers to three decimal places.) SST = SSR = SSE = (b) Compute the coefficient of determination r2. (Round your answer to three decimal places.) 2 = Comment on the goodness of fit. (For purposes of this exercise, consider a proportion large if it is at least 0.55.) 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. 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. O 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. O 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) What is the value of the sample correlation coefficient? (Round your answer to three decimal places.)

Sagot :

(a) SST = 1772,   SSR= 1420.79,  SSE=353.05

(b) Coefficient of determination is 0.8018.

(c) Sample correlation coefficient is 0.8954.

The average score can be obtained as

y = ∑y / n

  = 330 / 6

y = 55

The least-square regression line is given as:

y' = 23.221 + 0.318x

Using the above stated regression equation; the predicted values of overall score can be obtained for the given values of price.

y' = 23.321 + 0.318(170) = 77.281

(a) The formula for computing SST is given as:

SST = ∑ (yi - y)²

SST = 1772

Hence the value of SST is 1772.

The formula for computing SSR is given as:

SSR = ∑ (yi' - y)²

SSR = 1420.79

Hence the value of SSR is 1420.79.

The formula for computing SSE is given as:

SSE = ∑(yi - y')²

SSE = 353.05

(b) The coefficient of determination is given by the formula:

R² = [tex]\frac{SSR}{SST}[/tex]

   = [tex]\frac{1420.79}{1772}[/tex]

R²= 0.8954

Hence the value of the coefficient of determination is 0.8018.

(c) The correlation coefficient is computed as:

r = [tex]\sqrt{R^{2} }[/tex]

 = [tex]\sqrt{0.8018}[/tex]

r = 0.8954

Hence the value of the correlation coefficient is 0.8954.

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