STA_282_Ch_9_Sec_4_Notes.doc

9.4 Confidence Intervals For a Population Standard Deviation Objectives 1. Find critical values for the chi-square distribution. 2. Construct and interpret confidence intervals for the population variance and standard deviation. Chi-Square Distribution If a simple random sample of size n is obtained from a normally distributed population with mean µ and standard deviation ?, then ?2= ((n-1) * s2)/ ?2 has a chi-square distribution with n-1 degrees of freedom. Characteristics of the Chi-Square Distribution It is not symmetric. The shape of the chi-square distribution depends on the degrees of freedom, just like the Student?s t-distribution. As the number of degrees of freedom increases, the chi-square distribution becomes more nearly symmetric. The values of ?2 are nonnegative; that is, values of ?2 are always greater than or equal to 0. A (1-?)*100% Confidence Interval about ?2 If a simple random sample of size n is taken from a normal population with mean µ and standard deviation ?, then a (1-?)*100% confidence interval about ?2 is given by Lower bound: ( (n-1) * s2) / (?2 ?/2) Upper Bound: ( (n-1) * s2) / (?2 1-?/2) Caution: A confidence interval about the population variance or standard deviation is not of the form ?point estimate INCLUDEPICTURE "http://upload.wikimedia.org/math/8/f/c/8fc27b209ba0923aa52897d4cb0a3656.png" \* MERGEFORMATINET margin of error? because the sampling distribution of the sample variance is not symmetric.