Hw12Sol - SIEO 3600(IEOR Majors Solution of Assignment#12...

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Unformatted text preview: SIEO 3600 (IEOR Majors) Solution of Assignment #12 Introduction to Probability and Statistics April 27, 2010 Solution of Assignment #12 ( Regression: linear (simple and multiple) and polynomial ) 1. The following data represent the relationship between the number of alignment errors and the number of missing rivets for 10 different aircrafts. Number of Number of Missing Rivets = x Alignment Errors = y 13 7 15 7 10 5 22 12 30 15 7 2 25 13 16 9 20 11 15 8 (a) Plot a scatter diagram. (b) Estimate the regression coefficients. (c) Compute the coefficient of determination R 2 . Is a linear model a good fit? Solution: We compute from the given data: ¯ X ( n ) = 17 . 3 , ¯ Y ( n ) = 8 . 9 , n summationdisplay i =1 X i Y i = 1783 , n summationdisplay i =1 X 2 i = 3433 . Then we have ˆ β = ∑ n i =1 X i Y i- n ¯ X ( n ) ¯ Y ( n ) ∑ n i =1 X 2 i- n ¯ X ( n ) 2 = 1783- 10 · 17 . 3 · 8 . 9 3433- 10 · 17 . 3 2 = 0 . 5528 , ˆ α = ¯ Y ( n )- ˆ β ¯ X ( n ) = 8 . 9- . 5528 · 17 . 3 =- . 6639 , R 2 = 1- S RR S Y Y = 1- ∑ n i =1 ( Y i- ˆ α- ˆ β X i ) 2 ∑ n i =1 ( Y i- ¯ Y ( n )) 2 = 0 . 9863 . Since we have such a big R 2 , we can say the linear model is a good fit. 2. The following data indicate the gain in reading speed versus the number of weeks in the program of 10 students in a speed-reading problem. 2 SIEO 3600, Solution of Assignment #12 Number of weeks Speed Gain (wds/min) 2 21 3 42 8 102 11 130 4 52 5 57 9 105 7 85 5 62 7 90 (a) Plot a scatter diagram to see if a linear relationship is indicated. (b) Find the least squares estimates of the regression coefficients. (c) Compute the coefficient of determination R 2 . Is a linear model a good fit? Solution: We compute from the given data: ¯ X ( n ) = 6 . 1 , ¯ Y ( n ) = 74 . 6 , n summationdisplay i =1 X i Y i = 5387 , n summationdisplay i =1 X 2 i = 443 . Then we have ˆ β = ∑ n i =1 X i Y i- n ¯ X ( n ) ¯ Y ( n ) ∑ n i =1 X 2 i- n ¯ X ( n ) 2 = 5387- 10 · 6 . 1 · 74 . 6 443- 10 · 6 . 1 2 = 11 . 797 , ˆ α = ¯ Y ( n )- ˆ β ¯ X ( n ) = 74 . 6- 11 . 797 · 6 . 1 = 2 . 639 , R 2 = 1- S RR S Y Y = 1- ∑ n i =1 ( Y i- ˆ α- ˆ β X i ) 2 ∑ n i =1 ( Y i- ¯ Y ( n )) 2 = 0 . 9863 ....
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Hw12Sol - SIEO 3600(IEOR Majors Solution of Assignment#12...

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