lecture06 notes

Probability and Statistics for Engineering and the Sciences (with CD-ROM and InfoTrac )

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Stat 312: Lecture 06 Confidence Intervals Moo K. Chung mchung@stat.wisc.edu 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) [1] 30.18341 47.53799 > c(mean(a3)-b,mean(a3)+b) [1] 32.18435 49.53893 > a4<-rnorm(10,42,14) > c(mean(a4)-b,mean(a4)+b) [1] 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 [1] 8.67729 > a1<-rnorm(10,42,14) > a1 [1] 72.97298 39.29226 48.76871 52. > c(mean(a1)-b,mean(a1)+b) [1] 38.63614 55.99072 ... If you this many many times .... > a17<-rnorm(10,42,14) > c(mean(a17)-b,mean(a17)+b) [1] 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 ...
View Full Document

Ask a homework question - tutors are online