Math Problem Statement

The following data show the brand, price (S), and the overall score for six stereo headphones that were tested by Consumer Reports (Consumer Reports website). 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 ý = 20.6485 + 0.3435г , where I = price (S) and y = overall score. Brand Bose Scullcandy Koss Philips/O’Neil Denon IVC Price ($) 170 150 95 70 60 55 Score 76 71 62 56 30 35 a. Compute SST, SSR, and SSE (to 3 decimals). SST = SSR = SSE= b. Compute the coefficient of determination r? (to 3 decimals). Y3= Comment on the goodness of fit. Hint: If 7' is greater than 0.70, the estimated regression equation provides a good fit. The least squares line - Select your answer - V a good fit as - Select your answer - proportion of the variability in y has been explained by the least squares line. c. What is the value of the sample correlation coefficient (to 3 decimals)? Txy Just give answers

Solution

Answers:

a. Compute SST, SSR, and SSE:

  • SST = 1357.333
  • SSR = 1085.333
  • SSE = 272.000

b. Compute the coefficient of determination (r2r^2):

  • r2=0.799r^2 = 0.799

Comment: The regression line is a good fit as r2r^2 is greater than 0.70, indicating that 79.9% of the variability in yy is explained by the regression line.

c. Value of the sample correlation coefficient (rr):

  • r=0.894r = 0.894

Would you like detailed calculations or clarifications?

Related Questions:

  1. How are SST, SSR, and SSE calculated in regression analysis?
  2. Why does a higher r2r^2 value indicate a better fit?
  3. How does the correlation coefficient relate to r2r^2?
  4. What are other statistical methods to measure goodness-of-fit?
  5. How does the price affect the score based on this regression analysis?

Tip: The square of the correlation coefficient (r2r^2) always equals the coefficient of determination.

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Math Problem Analysis

Mathematical Concepts

Linear Regression
Statistical Analysis
Coefficient of Determination
Correlation Coefficient

Formulas

SST = Σ(y_i - ŷ)^2
SSR = Σ(ŷ - ȳ)^2
SSE = Σ(y_i - ŷ)^2
r² = SSR / SST
r = √r²

Theorems

Least Squares Regression
Coefficient of Determination
Sample Correlation Coefficient

Suitable Grade Level

Grades 11-12

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