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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  tn1,1 , 2 n s x + tn1,1 2 n .
iid where s is the sample standard deviation and tn1,1 is the (1  )th percentile of a tn1 distri2 2 bution i.e., the (1  )th percentile of a tdistribution with n  1 degrees of freedom. 2 To find what tn1,1 is in R, type: 2 qt(1alpha/2,df=n1) where you replace n and alpha with the appropriate numbers! 1. 90% CI: = 0.10, (1  ) = 0.950, tn1,0.950 =qt(0.950,df=n1); 2 2. 95% CI: = 0.05, (1  ) = 0.975, tn1,0.975 =qt(0.975,df=n1); 2 3. 99% CI: = 0.01, (1  ) = 0.995, tn1,0.995 =qt(0.995,df=n1); 2 Poisson CI Let Xi Poisson () for i = 1, . . . , n. Let x be the sample mean for a given dataset, then the normalapproximation 100(1  )% CI for : x x x  tn1,1 , x + tn1,1 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  tn1,1/2 ^ p(1  p) ^ ^ , p + tn1,1/2 ^ 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.
 Fall '10
 DRAKE
 Normal Distribution

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