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Unformatted text preview: Stat 312: Lecture 06 Confidence Intervals Moo K. Chung firstname.lastname@example.org September 21, 2004 1. A confidence interval (CI) is an interval used to estimate the likely size of a population parameter. A confidence level is a measure of the degree of reliability of the confidence interval. Most commonly used confidence levels are the 90%, 95% and 99% confidence intervals that have 0.90, 0.95 and 0.99 probabilities respectively of containing the parameter. Ex. For population parameter , 95% confidence interval (^L , U ) of is an interval ^ that satisfies P (^L U ) = 0.95. ^ We usually make the interval centered so that P (^L ) = P ( U ) = 0.025. ^ 2. Let Xi N (, 2 ) with known 2 and unknown . 95% confidence interval for is L = x-1.96/ n, U = x+1.96/ n. ^ ^ 3. Let Xi N (, 2 ) with known 2 and unknown . 100(1 - )% confidence interval for is. L = x -z/2 / n, U = x +z/2 / n, ^ ^ where quantile z/2 is given by P Z > z/2 = . 2 > a2<-rnorm(10,42,14) > c(mean(a2)-b,mean(a2)+b)  30.18341 47.53799 > c(mean(a3)-b,mean(a3)+b)  32.18435 49.53893 > a4<-rnorm(10,42,14) > c(mean(a4)-b,mean(a4)+b)  40.65048 58.00506 in the long run. The following simulation demonstrates this. Let Xi N (, 142 ). Suppose = 42 but assume we do not know this fact. Let n = 10. This is the binge drinking example of Lecture 1. > b=1.96*14/sqrt(10) > b  8.67729 > a1<-rnorm(10,42,14) > a1  72.97298 39.29226 48.76871 52. > c(mean(a1)-b,mean(a1)+b)  38.63614 55.99072 ... If you this many many times .... > a17<-rnorm(10,42,14) > c(mean(a17)-b,mean(a17)+b)  45.15059 62.50517 Review problems. Example 7.2.,7.3. 4. A 95% confidence interval can be interpreted probabilistically as an interval that can contain true unknown parameter 95% of time ...
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- Fall '04