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Formula_Sheet

# Formula_Sheet - STA 100 Confidence Interval Formulae Normal...

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STA 100: Confidence Interval Formulae Normal CI for μ with known σ : If σ is known and X i iid N ( μ, σ 2 ) for i = 1 , . . . , n , then a 100(1 - α )% CI for μ is given by: ¯ x - z 1 - α 2 · σ n , ¯ x + z 1 - α 2 · σ n . where z 1 - α 2 is the (1 - α 2 )th percentile of a normal distribution. Depending on what level confidence interval you want. . . 1. 90% CI: α = 0 . 10, (1 - α 2 ) = 0 . 950, z 0 . 950 = 1 . 64; 2. 95% CI: α = 0 . 05, (1 - α 2 ) = 0 . 975, z 0 . 975 = 1 . 96; 3. 99% CI: α = 0 . 01, (1 - α 2 ) = 0 . 995, z 0 . 995 = 2 . 58. Normal CI for μ with unknown σ If X i iid N ( μ, σ 2 ) for i = 1 , . . . , n , and σ is unknown then a 100(1 - α )% CI for μ is given by: ¯ x - t n - 1 , 1 - α 2 · s n , ¯ x + t n - 1 , 1 - α 2 · s n . where s is the sample standard deviation and t n - 1 , 1 - α 2 is the (1 - α 2 )th percentile of a t n - 1 distri- bution i.e., the (1 - α 2 )th percentile of a t - distribution with n - 1 degrees of freedom. To find what t n - 1 , 1 - α 2 is in R , type: qt(1-alpha/2,df=n-1) where you replace n and alpha with the appropriate numbers! 1. 90% CI: α = 0 . 10, (1 - α 2 ) = 0 . 950, t n - 1 , 0 . 950 = qt(0.950,df=n-1) ; 2. 95% CI: α = 0 . 05, (1 - α 2 ) = 0 . 975, t n - 1 , 0 . 975 = qt(0.975,df=n-1)
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