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Formula_Sheet - STA 100: Confidence Interval Formulae...

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Unformatted text preview: STA 100: Confidence Interval Formulae Normal CI for with known : If is known and Xi N (, 2 ) for i = 1, . . . , n, then a 100(1 - )% CI for is given by: x - z1- , 2 n x + z1- 2 n . iid where z1- is the (1- )th percentile of a normal distribution. Depending on what level confidence 2 2 interval you want. . . 1. 90% CI: = 0.10, (1 - ) = 0.950, z0.950 = 1.64; 2 2. 95% CI: = 0.05, (1 - ) = 0.975, z0.975 = 1.96; 2 3. 99% CI: = 0.01, (1 - ) = 0.995, z0.995 = 2.58. 2 Normal CI for with unknown If Xi N (, 2 ) for i = 1, . . . , n, and is unknown then a 100(1 - )% CI for is given by: s x - tn-1,1- , 2 n s x + tn-1,1- 2 n . iid where s is the sample standard deviation and tn-1,1- is the (1 - )th percentile of a tn-1 distri2 2 bution i.e., the (1 - )th percentile of a t-distribution with n - 1 degrees of freedom. 2 To find what tn-1,1- is in R, type: 2 qt(1-alpha/2,df=n-1) where you replace n and alpha with the appropriate numbers! 1. 90% CI: = 0.10, (1 - ) = 0.950, tn-1,0.950 =qt(0.950,df=n-1); 2 2. 95% CI: = 0.05, (1 - ) = 0.975, tn-1,0.975 =qt(0.975,df=n-1); 2 3. 99% CI: = 0.01, (1 - ) = 0.995, tn-1,0.995 =qt(0.995,df=n-1); 2 Poisson CI Let Xi Poisson () for i = 1, . . . , n. Let x be the sample mean for a given dataset, then the normal-approximation 100(1 - )% CI for : x x x - tn-1,1- , x + tn-1,1- 2 2 n n iid Remember: this uses the CLT, so it needs large enough n! Rule of thumb: n > 60 and n > 30. x Binomial CI Let X Bin (n, p). If we observe X = x, then let your estimate of p be p = n . ^ x An approximate 100(1 - )% CI for p is given by: p - tn-1,1-/2 ^ p(1 - p) ^ ^ , p + tn-1,1-/2 ^ n p(1 - p) ^ ^ n . Where the approximation (which comes from the CLT) is only good if n^ > 5 and n(1 - p) > 5. p ^ ...
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This note was uploaded on 01/17/2012 for the course STAT 100 taught by Professor drake during the Fall '10 term at UC Davis.

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