This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.View Full Document
Unformatted text preview: M4056 Confidence Intervals November 15, 2010 Let be a parameter of a probability distribution. A confidence interval for is a random interval, calculated from a sample, that contains with some specified probability. (A random interval is an interval [ L ( vector X ) , U ( vector X )], where L and U are statistics.) Example 1. Consider a sample X of size 1 an from a normal population with (unknown) mean and (known) variance 2 . Let us find a 95% confidence interval for . This is easily done by normalizing X . [ X r, X + r ] X [ r, + r ] X [ r, r ] X [ r/, r/ ] Z [ r/, r/ ] , Z standard normal . Now, P ( Z [ 2 , 2]) = 0 . 9545 . . . , so if we set r = 2 we get the desired interval. Example 1, continued. There is a direct connection to hypothesis testing. Let H be the hypothesis that = . Suppose we wish to test this against the alternative H 1 : negationslash = , and we desire a test of level 0...
View Full Document