Section 7.4: Confidence Interval for Varianceand Standard Deviation of a Normal Population1Suppose X1, ···, Xnare iid observations of a nor-mal sample, say N(µ, σ2). Then,(n − 1)S2σ2∼ χ2n−1.Thus, we haveP (χ21−α/2,n−1≤(n − 1)S2σ2≤ χ2α/2,n−1) = 1 − αwhich gives the 1 −α level confidence interval forσ2as[(n − 1)s2χ2α/2,n−1,(n − 1)s2χ21−α/2,n−1].and for σ as[vuuut(n − 1)s2χ2α/2,n−1,vuuut(n − 1)s2χ21−α/2,n−1].2First example of Section 7.4: example 7.15 ontextbook. In this example, the data collected 17observations of breakdown voltage. We have n =17 and s2= 137324.3. Thus for 95% confidenceinterval for σ2, we needχ20.975,16= 6.908andχ20.025,16= 28.845.Thus, the 95% confidence interval for σ2is[16s228.845,16s26.908] = [76172.3, 318064.4]and the 95% confidence interval for σ is[√76172.3,√318064.4] = [276.0,
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