Lesson_10_11_Review_Problem_Solutions_F12

y 11524 247x excel

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Unformatted text preview: the x value moves away from the mean. PROBLEM # 10.21 The coefficient of correlation (r) describes both the direction and the strength of the linear relationship between two variables. The coefficient of determination (r2) expresses the proportion of the variation in the dependent variable (y) that is explained by the regression ˆ line, y = b0 + b1x, but it does not indicate the direction of the relationship. PROBLEM # 10.22 Scatter Diagram 20 Sales 15 10 5 0 0 20 40 60 80 100 120 Advertising b x i y i x i2 23 9.6 529 92.16 220.8 46 11.3 2,116 127.69 519.8 60 12.8 3,600 163.84 768.0 54 yi2 9.8 2,916 96.04 529.2 x i yi 5 28 8.9 784 79.21 249.2 33 12.5 1,089 156.25 412.5 25 12.0 625 144.00 300.0 31 11.4 961 129.96 353.4 36 12.6 1,296 158.76 453.6 88 13.7 7,744 187.69 1205.6 90 14.4 8,100 207.36 1296.0 99 15.9 9,801 252.81 1,574.1 Total 613 7,882.2 144.9 n n ∑ x i = 613 s xy Ⱥ Ⱥ n 1 Ⱥ = x i yi − n − 1 Ⱥ i =1 Ⱥ Ⱥ Ⱥ ∑ i =1 y i = 144.9 i =1 ∑ n ∑ x i2 = 39,561 i =1 n n ∑ ∑y xi i =1 i i =1 n 39,561 n ∑x y i i 1,795.77 = 7,882.2 i =1 Ⱥ Ⱥ 1 Ⱥ (613)(144.9 Ⱥ Ⱥ Ⱥ = 12 − 1 Ⱥ7,882.2 − Ⱥ = 43.66 12 Ⱥ Ⱥ Ⱥ Ⱥ Ⱥ 2 Ⱥ ȹ n ȹ Ⱥ Ⱥ ȹ ȹ Ⱥ xi n ȹ ȹ Ⱥ 1 Ⱥ (613) 2 Ⱥ 1 Ⱥ ȹ i =1 Ⱥ Ⱥ 2 2 Ⱥ x i − sx = = Ⱥ39,561 − Ⱥ = 749.7 n − 1 Ⱥ i =1 n Ⱥ 12 − 1 Ⱥ 12 Ⱥ Ⱥ Ⱥ Ⱥ Ⱥ Ⱥ Ⱥ Ⱥ Ⱥ ∑ ∑ b1 = s xy = s2 x x= ∑x y= n i = 43.66 = .0582 749.7 613 = 51.08 12 ∑y n i = 144.9 = 12.08 12 b 0 = y − b1x = 12.08 – (.0582)(51.08) = 9.107 The sample regression line is 6 ˆ y = 9.107 + .0582x The slope tells us that for each additional thousand dollars of advertising sales increase on average by .0582 million. The y ­intercept has no practical meaning. PROBLEM # 10.23 Determine the standard error of estimate a. b. c. d. Is there evidence of a linear relationship between advertising and sales? Estimate ! 1 with 95% confidence. Compute the coefficient of determination and interpret the value. Briefly summarize what you have learned in parts c,d,e, and f. 2 Ⱥ ȹ n ȹ Ⱥ Ⱥ ȹ ȹ Ⱥ yi n ȹ ȹ Ⱥ (144.9) 2 Ⱥ 1 Ⱥ 1 Ⱥ ȹ i =1 Ⱥ Ⱥ 2 2 Ⱥ y i − sy = 1,795.77 − = Ⱥ Ⱥ = 4.191 n − 1 Ⱥ i =1 n 12 Ⱥ Ⱥ 12 − 1 Ⱥ Ⱥ Ⱥ Ⱥ Ⱥ Ⱥ Ⱥ Ⱥ Ⱥ ∑ ∑ ȹ ȹ s 2 ȹ (43.66) 2 xy SSE = (n − 1)ȹ s 2 − 2 ȹ = (12 − 1)ȹ 4.191 − ȹ ȹ y s x ȹ 749.7 ȹ ȹ Ⱥ sε = ȹ ȹ = 18.13 ȹ Ⱥ 18.13 SSE = = 1.347 (Excel: s ε = 1.347) n−2 12 − 2 d H 0 : β1 = 0 H1 : β1 ≠ 0 Rejection region: t &gt...
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