# chapter7 - Chapter 7 Inferences Regarding Population...

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Unformatted text preview: Chapter 7 Inferences Regarding Population Variances Introduction • Population Variance : Measure of average squared deviation of individual measurements around the mean • Sample Variance : Measure of “average” squared deviation of a sample of measurements around their sample mean. Unbiased estimator of σ 2 N Y Y E Y V N i i ∑ =- =- = = 1 2 2 2 ) ( ] ) [( ) ( μ μ σ ( 29 1 1 2 2-- = ∑ = n y y s n i i Sampling Distribution of s 2 (Normal Data) • Population variance ( σ 2 ) is a fixed (unknown) parameter based on the population of measurements • Sample variance ( s 2 ) varies from sample to sample (just as sample mean does) • When Y ~ N ( μ , σ ), the distribution of (a multiple of) s 2 is Chi-Square with n-1 degrees of freedom . ( n-1) s 2 / σ 2 ~ χ 2 with df= n-1 • Chi-Square distributions – Positively skewed with positive density over (0, ∞ ) – Indexed by its degrees of freedom (df) – Mean=df, Variance=2(df) – Critical Values given in Table 8, pp. 686-687 Chi-Square Distributions 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 10 20 30 40 50 60 70 f(X^2) X^2 Chi-Square Distributions f1(y) f2(y) f3(y) f4(y) f5(y) df=4 df=10 df=20 df=30 df=50 Chi-Square Distribution Critical Values-0.02 0.02 0.04 0.06 0.08 0.1 0.12 5 10 15 20 25 30 35 40 f(X^2) X^2 Chi-Square Distribution (df=10) .95 .025 .025 3.247 20.48 α χ 2( α ) df=10 0.995 2.156 0.990 2.558 0.975 3.247 0.950 3.940 0.900 4.865 0.100 15.987 0.050 18.307 0.025 20.483 0.010 23.209 0.005 25.188 Chi-Square Critical Values (2-Sided Tests/CIs) f(X^2) 1- α α /2 α /2 χ 2 L χ 2 U (1- α )100% Confidence Interval for σ 2...
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chapter7 - Chapter 7 Inferences Regarding Population...

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