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**Unformatted text preview: **Chapter 4 Simultaneous inferences and other topics in Regression analysis 4.1 Bonferroni joint CIs for and 1 We will learn a procedure for constructing simultaneous CIs for and 1 with a specified family confidence coefficient . Note that the meaning of family confidence coefficient , say .95, is that if repeated samples are selected and interval estimate for both and 1 are calculated for each sample by specified procedure, 95% of the samples would lead to a family of estimates where both CI are correct; for 5% of the samples, either one or both of the interval estimates would be incorrect. Consider a multiple hypotheses. If each test in a family of g tests is tested at level of significance , the probability making one or more Type I errors if all g hypotheses are true is * where: g ) 1 ( 1 * (Bonferroni inequality) Equality holds if the tests are mutually independent. It is apparent that 1 ) 1 ( 1 * g g That is, the probability is close to 1 that we falsely reject at least one hypothesis even if they are all true. If is small then g g ) 1 ( 1 * . Hence, if we perform each of g tests at the g / level we get: g g 1 1 This approach to multiple testing is referred to as the Bonferroni method or procedure . So, using the Bonferroni procedure, we perform each of g tests at the g / level and are assured that the probability is of making one or more Type I errors if all g hypotheses are true. For example, if we perform 4 tests and use the 0.05/4 = .0125 level of significance for each test, the probability we falsely reject one or more of the 4 corresponding hypotheses is less than or equal to 0.05. In the simple linear regression model, simultaneous (Bonferroni) confidence limits for estimating and 1 are: } { b Bs b and } { 1 1 b Bs b where ) 4 / 1 ( )) 2 /( 1 ( 2 2 n n t g t B Example: Toluca Company example We are interested in finding Bonferroni joint CIs for and 1 : To estimate 90% CIs for and 1 , we need . 069 . 2 ) 975 (. ) 4 / 1 . 1 ( ) 4 / 1 ( 23 2 25 2 t t t B n From The MEANS Procedure Variable Sum cross 70690.00 xdev2 19800.00 The MEANS Procedure Analysis Variable : lotsize N Mean Std Dev Minimum Maximum...

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