Asked by Lemy
Analyzing Regression Results
LaTasha was presented with the following data set and argued that there was no correlation between x and y. Is LaTasha correct? Use the regression equation to explain your reasoning.
A 2-row table with 7 columns. The first row is labeled x with entries 1, 2, 3, 4, 5, 6, 7. The second row is labeled y with entries 4, 5, 4, 5, 4, 5, 4.
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Answered by
GPT-5 mini
AI
Compute the least-squares line. x̄ = (1+...+7)/7 = 4. ȳ = (4+5+4+5+4+5+4)/7 = 31/7 ≈ 4.42857.
Sxx = Σ(xi − x̄)² = 9+4+1+0+1+4+9 = 28.
Sxy = Σ(xi − x̄)(yi − ȳ) = (−3)(−3/7)+(−2)(4/7)+(−1)(−3/7)+0(4/7)+1(−3/7)+2(4/7)+3(−3/7) = 0.
Slope b = Sxy/Sxx = 0/28 = 0. Intercept a = ȳ − b x̄ = 31/7. So the regression equation is
ŷ = 31/7 ≈ 4.4286.
A slope of 0 means there is no linear relationship between x and y (the Pearson r is 0). So LaTasha is correct that there is no correlation in the linear sense. (The data do have a nonrandom alternating pattern, but it produces no linear trend.)
Sxx = Σ(xi − x̄)² = 9+4+1+0+1+4+9 = 28.
Sxy = Σ(xi − x̄)(yi − ȳ) = (−3)(−3/7)+(−2)(4/7)+(−1)(−3/7)+0(4/7)+1(−3/7)+2(4/7)+3(−3/7) = 0.
Slope b = Sxy/Sxx = 0/28 = 0. Intercept a = ȳ − b x̄ = 31/7. So the regression equation is
ŷ = 31/7 ≈ 4.4286.
A slope of 0 means there is no linear relationship between x and y (the Pearson r is 0). So LaTasha is correct that there is no correlation in the linear sense. (The data do have a nonrandom alternating pattern, but it produces no linear trend.)
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