Explanation Given info: The data shows that the life tests of cutting tools with the dependent variable ( y ) as useful life, and the independent variables ( x ) as cutting speed. Calculations : For Brand A, The regression line is obtained below, y = β 0 + β 1 x The table represents the measurements, Cutting Speed( x) Useful Life( y ) x 2 y 2 x y 30 4.5 900 20.25 135 30 3.5 900 12.25 105 30 5.2 900 27.04 156 40 5.2 1,600 27.04 208 40 4.0 1,600 16 160 40 2.5 1,600 6.25 100 50 4.4 2,500 19.36 220 50 2.8 2,500 7.84 140 50 1.0 2,500 1 50 60 4.0 3,600 16 240 60 2.0 3,600 4 120 60 1.1 3,600 1.21 66 70 1.1 4,900 1.21 77 70 0.5 4,900 0.25 35 70 3.0 4,900 9 210 ∑ x = 750 ∑ y = 44.8 ∑ x 2 = 40 , 500 ∑ y 2 = 168.7 ∑ x y = 2 , 022 Formula to compute the estimates of β 0 and β 1 is, β ^ 1 = S S x y S S x x β ^ 0 = y ¯ − β ^ 1 x ¯ Where, S S x x = ∑ x i 2 − ( ∑ x i ) 2 n S S x y = ∑ x i y i − ∑ x i ∑ y i n x ¯ = ∑ i = 1 n y i n y ¯ = ∑ i = 1 n y i n Substitute the values, ∑ x = 750 , ∑ y = 44.8 , ∑ x 2 = 40 , 500 , ∑ y 2 = 168.7 and ∑ x y = 2 , 022 to obtain S S x y , S S x x , x ¯ , and y ¯ . The value of S S x x is, S S x x = ∑ x i 2 − ( ∑ x i ) 2 n = 40 , 500 − ( 750 ) 2 15 = 40 , 500 − 37 , 500 = 3 , 000 The value of S S x x is 3,000. The value of S S x y is, S S x y = 2 , 022 − ( 750 ) ( 44.8 ) 15 = 2 , 022 − 2240 = − 218 The value of S S x y is –218. The value of S S y y is, S S y y = ∑ y i 2 − ( ∑ y i ) 2 n = 168.70 − ( 44.8 ) 2 15 = 168.70 − 133.8026 = 34.8973 The value of S S y y is 34.8973. The value of x ¯ is, x ¯ = 750 15 = 50 The value of x ¯ is 50. The value of y ¯ is, y ¯ = 44.8 15 = 2.9866 The value of y ¯ is 2.9866. Substitute the value S S x y = − 218 , and S S x x = 3000 in β ^ 1 , The value of β ^ 1 is, β ^ 1 = S S x y S S x x = − 218 3000 = − 0.0726 Substitute the values y ¯ = 2.9866 , x ¯ = 50 , and β ^ 1 = − 0.0726 in β ^ 0 . The value of β ^ 0 is, β ^ 0 = 2.9866 − ( − 0.0726 ) ( 50 ) = 6.62 The value of β ^ 0 is 6.62. The fitted regression line is, y ^ = β 0 + β 1 x y ^ = 6.62 − 0.0276 x Therefore, the fitted regression line is y ^ = 6.62 − 0.0726 x . The formula for s is obtained below: The value of SSE is, S S E = S S y y − β ^ 1 S S x y = ( 34.8973 ) ( − 0.0276 ) ( − 218 ) = 19.0560 The value of SSE is 19.0560. Substitute 19.0560 for SSE and 15 for n in the formula. The value of s is, s = 19.0560 15 − 2 = 1.4658 = 1.211 Therefore, For Brand A the value of s is 1.211. For Brand B, The regression line is obtained below, y = β 0 + β 1 x The table represents the measurements, Cutting Speed( x) Useful Life( y ) x 2 y 2 x y 30 6 900 36 180 30 6

13th Edition

ISBN: 9780134080215

Author: James T. McClave, Terry T Sincich

Publisher: PEARSON

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Chapter 11.3, Problem 52ACA

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Which brand in using the least squares line to predict useful life for a cutting speed.

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Chapter 11.3, Problem 51ACI

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