# L21 - This following material covers Sec tion 4.4...

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Unformatted text preview: This following material covers Sec- tion 4.4: confidence intervals for proportions and variances. You are not responsible for the deriva- tion (shown in purple ) of these confidence intervals on a test or final exam. I have shown the details for those who are inter- ested. You should know how to use the tables, how to construct and interpret the confidence in- tervals, and the meanings of stan- dard error, margin of error. 1 Estimating a Variance Given . N ( μ,σ 2 ) population, random sample { X 1 ,...,X n } . Find C.I. on σ 2 Use fact that W = ( n- 1) S 2 σ 2 has a chi-square distribution on n- 1 degrees of freedom. Upper percentage points χ 2 ( A,n- 1) satisfy upper tail area condition P ( W > χ 2 ( A,n- 1)) = A Given by Table C5 p 570. For confidence in- tervals A will be α/ 2 where typically α = 0 . 05 but χ 2 ( A,n- 1) has the same meaning as an upper percentile of W for all values of A. 2 3 Deriving 100(1- α ) C.I. on σ 2 . Start with statement P ( χ 2 (1- α/ 2 ,n- 1) ≤ W ≤ χ 2 ( α/ 2 ,n- 1)) = 1- α Substitute W = ( n- 1) S 2 σ 2 and rearrange. Get P ( n- 1) S 2 χ 2 ( α/ 2 ,n- 1) ≤ σ 2 ≤ ( n- 1) S 2 χ 2 (1- α/ 2 ,n- 1) ! = 1- α 100(1- α )% CI: ( n- 1) S 2 χ 2 ( α/ 2 ,n- 1) , ( n- 1) S 2 χ 2 (1- α/ 2 ,n- 1) ! To get C.I. on σ take square roots of end points. 4 Example 4.4-1 . n = 16 , ¯ x = 4 . 3 ,s = 0 . 6 . 95% CI on σ 2 is ( n- 1) S 2 χ 2 ( α/ 2 ,n- 1) , ( n- 1) S 2 χ 2 (1- α/ 2 ,n- 1) ! = 15(0 . 6) 2 χ 2 (0 . 025 , 15) , 15(0 . 6) 2 χ 2 (1- . 025 , 15) ! = 15(0 . 6) 2 χ 2 (0 . 025 , 15) , 15(0 . 6) 2 χ 2 (1- . 025 , 15) ! = 15(0 . 6) 2 27 . 488 , 15(0 . 6) 2 6 . 262 ! =(0 . 20 , . 86) 95% CI on σ 2 is ( √ . 20 , √ . 86) = (0 . 0447 , . 929) 5 Comparing Two Variances Independent samples of sizes n 1 ,n 2 from two normal populations, variances σ 1 ,σ 2 2 respectively.respectively....
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## This note was uploaded on 01/03/2012 for the course EE 1244 taught by Professor Drera during the Fall '10 term at Conestoga.

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L21 - This following material covers Sec tion 4.4...

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