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- Michigan
- Central Michigan University
- Statistics
- Statistics 282
- Wang
- STA_282_Ch_9_Sec_4_Notes.doc

Timothy B.

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 σ, then χ2= ((n-1) * s2)/ σ2 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 χ2 are nonnegative; that is, values of χ2 are always greater than or equal to 0. A (1-α)*100% Confidence Interval about σ2 If a simple random sample of size n is taken from a normal population with mean µ and standard deviation σ, then a (1-α)*100% confidence interval about σ2 is given by Lower bound: ( (n-1) * s2) / (χ2 α/2) Upper Bound: ( (n-1) * s2) / (χ2 1-α/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.

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