a. To determine Show the expression E ( σ ^ 2 ) = E ( M S E ) = σ 2 . Explanation Given info: The linear regression model is Y = β 0 + β 1 x + ε . E ( ε ) = 0 , V ( ε ) = σ 2 .the errors are uncorrelated. Calculation: Let, Y be the response variable and x be the predictor variable, then the simple linear regression model can be expressed as Y = β 0 + β 1 x + ε , where, β 0 and β 1 are the intercept and slope coefficient respectively. ε is the random error with mean zero and variance σ 2 The least square estimate of β 1 is defined as β ^ 1 = S x y S x x and the least square estimate of β 0 is defined as β ^ 0 = y ¯ − β ^ 1 x ¯ where, S x x = ∑ i = 1 n x i 2 − ( ∑ i = 1 n x i ) 2 n S x y = ∑ i = 1 n x i y i − ( ∑ i = 1 n x i ) ( ∑ i = 1 n y i ) n Proof: It is known that M S E = S S E n − 2 Where M S E , S S E are the mean error sum of squares and the error sum of squares respectively. S S E = ∑ i = 1 n e i 2 = ∑ i = 1 n ( y i − y ^ i ) 2 It is known that y ^ i = β ^ 0 + β ^ 1 x i = y ¯ − β ^ 1 x ¯ + β ^ 1 x i = y ¯ + β ^ 1 ( x i − x ¯ ) Hence, e i = ( y i − y ^ i ) = y ^ i − y ¯ + β ^ 1 ( x i − x ¯ ) Hence, E ( M S E ) = E ( S S E n − 2 ) = E ( S S E n − 2 ) = E ( ∑ i = 1 n e 2 i n − 2 ) = ∑ i = 1 n E ( e 2 i ) n − 2 It is known that V ( e i ) = E ( e 2 i ) − ( E ( e i ) ) 2 Now, taking expectation on both side of e i = ( y i − y ^ i ) , E ( e i ) = E ( y i − y ^ i ) = E ( y ^ i − y ¯ + β ^ 1 ( x i − x ¯ ) ) = E ( y ^ i ) − E ( y ¯ ) + ( x i − x ¯ ) E ( β ^ 1 ) = 0 Now, taking variance on both side of e i = ( y i − y ^ i ) , V ( e i ) = V ( y i ) + V ( y ^ i ) − 2 cov ( b. To determine Show the expression E ( M S R ) = σ 2 + β 2 1 S x x .

Applied Statistics and Probability for Engineers

6th Edition

ISBN: 9781118539712

Author: Douglas C. Montgomery

Publisher: WILEY

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A new framework of data analysis where the new statistical methods are used for interpreting the results and analyzing the data is known as estimation in statistics.

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Chapter 11, Problem 111SE

To determine

Show the expression E(σ^2)=E(MSE)=σ2.

To determine

Show the expression E(MSR)=σ2+β21Sxx.

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Chapter 11, Problem 110SE

Chapter 11, Problem 112SE

Chapter 11 Solutions

Applied Statistics and Probability for Engineers

Ch. 11.2 - 11-1. Diabetes and obesity are serious health...Ch. 11.2 - 11-2. On average, do people gain weight as they...Ch. 11.2 - Prob. 3E Ch. 11.2 - 11-4. Regression methods were used to analyze the...Ch. 11.2 - 11-5. See Table E11-1 for data on the ratings of...Ch. 11.2 - 11-6. An article in Technometrics by S. C. Narula...Ch. 11.2 - 11-7. The number of pounds of steam used per month...Ch. 11.2 - 11-8. Go Tutorial Table E11-3 presents the highway...Ch. 11.2 - Prob. 9E Ch. 11.2 - 11-10. An article in the Journal of Sound and...

Ch. 11.2 - Prob. 11E Ch. 11.2 - 11-12. An article in the Journal of Environmental...Ch. 11.2 - 11-13. A rocket motor is manufactured by bonding...Ch. 11.2 - 11-14. Go Tutorial An article in the Journal of...Ch. 11.2 - 11-15 An article in the Journal of the...Ch. 11.2 - 11-16. An article in Wood Science and Technology...Ch. 11.2 - Prob. 17E Ch. 11.2 - Prob. 18E Ch. 11.2 - Prob. 19E Ch. 11.2 - 11-20. Show that in a simple linear regression...Ch. 11.2 - 11-21. Consider the simple linear regression model...Ch. 11.2 - 11-22. Suppose that we wish to fit a regression...Ch. 11.4 - 11-23. Recall the regression of percent body fat...Ch. 11.4 - Prob. 24E Ch. 11.4 - 11-25. Suppose that in Exercise 11-24 weight is...Ch. 11.4 - 11-26. Consider the simple linear regression model...Ch. 11.4 - Prob. 27E Ch. 11.4 - Prob. 28E Ch. 11.4 - Prob. 29E Ch. 11.4 - Prob. 30E Ch. 11.4 - 11-31. Consider the National Football League data...Ch. 11.4 - Prob. 32E Ch. 11.4 - Prob. 33E Ch. 11.4 - Prob. 34E Ch. 11.4 - 11-35. Consider the data from Exercise 11-9 on y =...Ch. 11.4 - Prob. 36E Ch. 11.4 - 11-37. Consider the data from Exercise 11-13, on y...Ch. 11.4 - 11-38. Consider the data from Exercise 11-12 on y...Ch. 11.4 - Prob. 39E Ch. 11.4 - Prob. 40E Ch. 11.4 - Prob. 41E Ch. 11.4 - Prob. 42E Ch. 11.4 - Prob. 44E Ch. 11.6 - 11-45. Using the regression from Exercise...Ch. 11.6 - 11-46. Q Using the regression from Exercise...Ch. 11.6 - 11-47. Refer to the data in Exercise 11-3 on y =...Ch. 11.6 - 1-48. Exercise 11-4 presented data on roadway...Ch. 11.6 - 11-49. Refer to the NFL quarterback ratings data...Ch. 11.6 - Prob. 50E Ch. 11.6 - 11-51. Exercise 11-7 presented data on y = steam...Ch. 11.6 - 11-52. Exercise 11-8 presented gasoline mileage...Ch. 11.6 - Prob. 53E Ch. 11.6 - Prob. 54E Ch. 11.6 - Prob. 55E Ch. 11.6 - 11-56. Exercise 11-12 presented data on chloride...Ch. 11.6 - 11-57. Refer to the data in Exercise 11-13 on...Ch. 11.6 - Prob. 58E Ch. 11.6 - Prob. 59E Ch. 11.7 - 11-60. Consider the simple linear regression model...Ch. 11.7 - 11-61. Repeat Exercise 11-60 using an error term...Ch. 11.7 - 11-62. Refer to the compressive strength data in...Ch. 11.7 - 11-63. Refer to the NFL quarterback ratings data...Ch. 11.7 - 11-64. Refer to the data in Exercise 11-6 on...Ch. 11.7 - 11-65. Refer to the data in Exercise 11-7 on y =...Ch. 11.7 - 11-66. Refer to the gasoline mileage data in...Ch. 11.7 - Prob. 67E Ch. 11.7 - Prob. 68E Ch. 11.7 - 11-69. Refer to Exercise 11-12. which presented...Ch. 11.7 - Prob. 70E Ch. 11.7 - 11-71. Consider the rocket propellant data in...Ch. 11.7 - 11-72. Consider the data in Exercise 11-9 on y =...Ch. 11.7 - Prob. 73E Ch. 11.7 - Prob. 74E Ch. 11.7 - Prob. 75E Ch. 11.8 - 11-76. Suppose that data are obtained from 20...Ch. 11.8 - 11-77. Suppose that data are obtained from 20...Ch. 11.8 - Prob. 78E Ch. 11.8 - 11-79. A random sample of 50 observations was made...Ch. 11.8 - 11-80. The data in Table E11-6 gave x = the water...Ch. 11.8 - Prob. 81E Ch. 11.8 - 11-82. The weight and systolic blood pressure of...Ch. 11.8 - Prob. 83E Ch. 11.8 - Prob. 84E Ch. 11.8 - 11-85. Refer to the NFL quarterback ratings data...Ch. 11.8 - Prob. 86E Ch. 11.9 - Prob. 87E Ch. 11.9 - Prob. 88E Ch. 11.9 - 11-89. An electric utility is interested in...Ch. 11.10 - Prob. 90E Ch. 11.10 - 11-91. The compressive strength of an alloy...Ch. 11.10 - Prob. 92E Ch. 11.10 - Prob. 93E Ch. 11.10 - Prob. 94E Ch. 11.10 - 11-95 Consider the propellant data is Exercise...Ch. 11 - Prob. 96SE Ch. 11 - Prob. 97SE Ch. 11 - 11-98. The strength of paper used in the...Ch. 11 - Prob. 99SE Ch. 11 - Prob. 100SE Ch. 11 - Prob. 101SE Ch. 11 - Prob. 102SE Ch. 11 - 11-103. An article in the Journal of Applied...Ch. 11 - 11-104. Two different methods can be used for...Ch. 11 - 11-105. The grams of solids removed from a...Ch. 11 - Prob. 106SE Ch. 11 - 11-107. The data in Table E11-20 related diamond...Ch. 11 - Prob. 108SE Ch. 11 - Prob. 109SE Ch. 11 - Prob. 110SE Ch. 11 - 11-111. Consider the simple linear regression...Ch. 11 - Prob. 112SE Ch. 11 - Prob. 113SE Ch. 11 - Prob. 114SE Ch. 11 - Prob. 115SE Ch. 11 - Prob. 116SE

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