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# Lecture 11 - Change in Homework 2 Assigned Problems Chapter...

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Change in Homework 2 Assigned Problems: Chapter 8 problems will now be due with HW Assignment 3 on April 4, 2008. HW Assignment 2 due on March 4, 2008 will include only Chapter 5 and 7 problems. Exam 2, on Tuesday, March 4 will be limited to Chapters 5 and 7.

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Given a random sample of size n from a normal input population with parameters μ and σ , the random variable (n-1)s 2 / σ 2 follows a Chi-Square ( Χ 2 ) distribution with n-1 degrees of freedom.

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The Χ α , ν 2 chi-square critical value corresponds to the value of the variable such that α x100% of the area under the curve lies to the right when there are ν =n-1 degrees of freedom. The chi-square distribution is tabulated in the appendix on page 673.
It follows that: Χ 2 α ,n-1 > Χ 2 (1- α ),n-1

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Since, in effect Χ 2 =(n-1)s 2 / σ 2 , we can solve for σ 2 in this definition and obtain the 100(1- α )% confidence interval for the variance of a normal population with the lower and upper limits: [ (n-1)s 2 / Χ 2 α /2,n-1 , (n-1)s 2 / Χ 2 1- α /2,n-1 ] A confidence interval for the standard deviation has lower and upper limits that are the square roots of the corresponding limits on the variance.
( ) 2 2 2 1 ~ ns dist χ σ ( ) () 22 11 χσ σχ −− =⇒ =

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Problem 7-42 (page 280) Determining the Chi-Square critical values
7-42 0.1,15 0.1,25 0.01,25 22.307 .) .) 34.38 44.31 .) .) 46.93 .) 11.52 .) 10.52 a b c d e f χ = = =

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Problem 7-43 (page 280) Applying the Chi-Square distribution Problem 7-45 (page 280) Computing a confidence interval on σ Coding data to simplify calculations
7-43) ( ) ( ) 22 2 .05,10 2 0.95,10 11 a.) 95th percentile 18.31 b.) 5th percentile 3.94 ns χσ σχ χ −− =⇒= = = 2 1, 3.94 v α 307 . 18 2 , v

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c.) ( ) 2 Prob 10.98 36.78 22 v χ ≤= 2 22 , 975 . 0 98 . 10 2 22 , 025 . 0 78 . 36
2 χ d.) () 22 25 Prob 14.61 or 37.65 0.10 v χχ = <>

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7-45) () ( ) 22 0.005,21 0.995,21 11 ns χχ ⎡⎤ −− ⎢⎥ ⎣⎦ ( ) /2, 1 1 1 nn αα 2 2 2 given that we estimate 1 the CI on the population variances is given by: σ χ =
7-45) 2

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Lecture 11 - Change in Homework 2 Assigned Problems Chapter...

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