Inference about Variances Sampling Distribution of Variances: You want to know the variance of a population. But you realize the population is too large to actually compute the true variance. So you decide to observe a sample from the population and use the sample variance as an estimate of the population variance. You need the sampling distribution of the sample variance in order to make inference about the population variance. The population is normally distributed with mean µ and standard deviation σ. Let y denote an observation from the population. Draw sample of size n → . 12,,nyy yK Compute the sample variance 22()1iiyysn−=−∑. Then sampling distribution of (n-1)s2/σ2 is chi-square with n-1 degrees of freedom. 1Inference about Variances Use the table for the chi-square distribution or a computer program to get probabilities. Examples: Suppose X has chi-square distribution with df=4. Then P(X > 11.1433) = .025. Notation: 2~kXχ X is distributed chi-square with k degrees of freedom ()2,akPX aχ>= Examples: 2.025,411.143χ= 2.025,1020.48χ= 2.975,103.247χ= 2Inference about Variances 95% Confidence Interval for a Variance: 222.975 .0252(1).95nsPχχσ⎛⎞−<<=⎜⎟⎝⎠ 22222.025 .975(1) (1).95ns nsPσχχ⎛⎞−−<< =⎜⎟⎝⎠ 95% confidence interval for σ2: 2222.025 .975(1) (1),nsnsχχ⎛⎞−−⎜⎟⎝⎠ Test of H0:σ2= σ02 versus Ha:σ2> σ02 Test statistic X = (n-1)s2/ σ02 Large values are “significant” 3Inference about Variances Example: Construct 95% confidence interval for σ2 for the hand-span for women The summary statistics for the hand-span data are my = 8.99 fy = 8.27 sm2 = 0.470 sf2 = 0.682 nm = 22 nf = 13 2222 2 2.025 .975 12,.025 12,.975( 1) ( 1) (13 1).682 (13 1).682,,8.184 8.184, (0.351,1.858)23.34 4.404nsnsχχ χ χ⎛⎞⎛⎞−− − −=⎜⎟⎜⎟⎜⎟⎝⎠⎝⎠⎛⎞==⎜⎟⎝⎠ 4Inference about Variances Use the F distribution to compare two variances. Reading probabilities from F table: The F distribution has numerator and denominator degrees of freedom F~Fdf1,df2 means F has df1 numerator and df2 denominator degrees of freedom ,1,2()adf dfPF F a>= Example: .05,12,212.25F = Property of F distribution: ,2,1(1/ )adf dfPF F a<= Result: 1,1,2 ,2,11/a df df a df dfFF−= Example: .95,21,121/ 2.25 0.44F == 5Inference about Variances Test of hypothesis Test of H0:σ1 2= σ22 versus Ha:σ1 2> σ22 Test statistic F = s1 2/s22 Large values are “significant” Example: Test H0:σm 2= σf2 versus Ha:σm 2> σf2 F = .470/.682=.689 <1 implies non-significant 6Inference about Variances Test of hypothesis Test of H0:σ1 2= σ22 versus Ha:σ1 2 ≠ σ22 Test statistic F = max(s1 2,s22)/min(s1 2,s22) Large values are “significant” Example: Test H0:σm 2= σf2 versus Ha:σm 2≠ σf2 F = .682 /.470 = 1.45, df1 = 12, df2 = 21 F12,21,.25 = 1.38, F12,21,.10 = 1.87 → .10<p<.25
View Full Document
Unlocking...