Confidence Interval Summary

Confidence Interval Summary - Stat 200 -Testing Summary...

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Unformatted text preview: Stat 200 -Testing Summary Page 1 Mathematicians are like Frenchmen; whatever you say to them, they translate it into their own language and forthwith it is something entirely different. Goethe 1. Large Sample Confidence Intervals • 100(1- α )% large-sample confidence interval for a population mean μ . X- z α 2 σ √ n , X + z α 2 σ √ n , where z α 2 is the constant such that P ( | Z | ≤ z α 2 ) = 1- α . Notes (a) The values of z α 2 for various confidence coefficients are readily obtained from the table of normal probabilities. In particular, 100(1- α )% z α 2 90% 1.645 95% 1.96 99% 2.58 (b) When σ is unknown, and n is large (say, bigger than 30), then it is usual to replace σ in the confidence interval formula by its estimator s . (c) The confidence interval result holds as long as X i have finite variance, not just when the X i ’s are normal. If, however, X i is normal, i = 1 ,...,n , then the confidence interval above is exact (i.e. has exact confidence coefficient 100(1- α )%). • 100(1- α )% large sample confidence interval for a difference in means μ X- μ Y . ( X- Y )- z α 2 s σ 2 X n 1 + σ 2 Y n 2 , ( X- Y ) + z α 2 s σ 2 X n 1 + σ 2 Y n 2 . When the true variances σ 2 X and σ 2 Y are unknown and n 1 and n 2 are large, then s 2 X and s 2 Y are used in place of σ 2 X and σ 2 Y , respectively, in the above formula. • 100(1- α )% large sample confidence interval for a proportion p . ˆ p- z α 2 r ˆ p (1- ˆ p ) n , ˆ p + z α 2 r ˆ p (1- ˆ p ) n . • 100(1- α )% large sample confidence interval for a difference in proportions p 1- p 2 . (ˆ p 1- ˆ p 2 )- z α 2 s ˆ p 1 (1- ˆ p 1 ) n 1 + ˆ p 2 (1- ˆ p 2 ) n 2 , (ˆ p 1- ˆ p 2 ) + z α 2 s ˆ p 1 (1- ˆ p 1 ) n 1 + ˆ p 2 (1- ˆ p 2 ) n 2 . 2. Large sample hypothesis tests Stat 200 -Testing Summary Page 2 Note: one-sided tests are given in parentheses ( ) and square brackets [ ]. These tests are valid for large n (say, n > 30), and population variances can be replaced by sample variances wherever appropriate. • Large sample hypothesis test for a population mean μ . We wish to test H : μ = μ against the alternative H A : μ 6 = μ ( H A : μ > μ ) [ H A : μ < μ ]. The test statistic is Z = X- μ σ/ √ n ....
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This note was uploaded on 02/07/2010 for the course STATS 620-202 taught by Professor R during the Two '09 term at University of Melbourne.

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Confidence Interval Summary - Stat 200 -Testing Summary...

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