Chapter 7 Inference about Variances Sampling Distribution of Variances Say you want to know the variance of a population What sample quantity have we seen before that might be a good estimator of the true population variance 2 We need the sampling distribution of the sample variance s2 in order to make inference about the population variance Assume the population is normally distributed with mean and standard deviation Let y denote an observation from the population 2 Recall that the sample variance is given by s 2 y i y 2 i n 1 2 Then sampling distribution of n 1 s is given by n2 1 or chi square with n 1 degrees of freedom We can use this distribution to model the sampling distribution of s2 and hence to construct confidence intervals and conduct hypothesis test The Chi square Distribution Usually we use a table for the chi square distribution or a computer program to get the probabilities we need for inference Examples Suppose has chi square distribution with df 4 Then P 11 1433 025 In general as with the normal distribution we look up values in the table for P X 2 k 1 Examples 2 025 4 11 143 2 025 10 20 48 2 975 10 3 247 Hence a 95 Confidence Interval for a Variance is given by 2 n 1 s 2 2 025 P 975 95 2 n 1 s 2 n 1 s 2 2 P 95 2 2 975 025 95 confidence interval for 2 n 1 s 2 n 1 s 2 2 2 975 025 Inference for the Variance of One Population Test of H0 2 02 vs Ha 2 02 Test statistic n 1 s2 02 Large values are significant Example Consider a data set that consists of the hand spans of men and women The summary statistics are given below Construct 95 confidence interval for 2 for the hand span for women 2 The summary statistics for the hand span data are ym 8 99 y f 8 27 sm2 sf2 0 682 nf 13 0 470 nm 22 n 1 s 2 n 1 s 2 13 1 682 13 1 682 2 2 2 2 975 12 975 025 12 025 8 184 8 184 0 351 1 858 23 34 4 404 Inference for Comparing the Variances of Two Populations Suppose we would like to compare the variances of two populations We again 2 2 use the sample estimate from each population Because the test statistic F s1 s2 has a known distribution we can use this distribution to conduct inference This distribution is called the F distribution Again we can read probabilities from an F table The F distribution has numerator and denominator degrees of freedom We write F Fdf1 df2 This means that F has df1 numerator and df2 denominator degrees of freedom and P F F df 1 df 2 Example F 05 12 21 2 25 An Important Property of the F distribution P F 1 F df 2 df 1 P F F df 1 df 2 Result F1 df 1 df 2 1 F df 2 df 1 3 Example F 95 21 12 1 2 25 0 44 Hypothesis Tests for Variances from Two Populations Hypothesis Test of H0 1 2 22 vs Ha 1 2 22 2 2 Test statistic F s1 s2 Large values are significant Example Test H0 m 2 f2 versus Ha m 2 f2 F 470 682 689 1 implies non significant Hypothesis Test of H0 1 2 22 versus Ha 1 2 22 Test statistic F max s1 2 s22 min s1 2 s22 Large values are significant Example Test H0 m 2 f2 versus Ha m 2 f2 F 682 470 1 45 df1 12 df2 21 F12 21 25 1 38 F12 21 10 1 87 10 p 25 4