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  2

Sampling Distribution of s 2 (Normal Data) Population variance (  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( ,  ), the distribution of (a multiple of) s 2 is Chi-Square with n-1 degrees of freedom. (n-1)s 2 /  2 ~  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

Chi-Square Distributions

Chi-Square Distribution Critical Values

Chi-Square Critical Values (2-Sided Tests/CIs) 2L2L 2U2U

(1-  )100% Confidence Interval for  2 (or  ) Step 1: Obtain a random sample of n items from the population, and compute s 2 Step 2: Choose confidence level (1-  ) Step 3: Obtain  2 L and  2 U from the table of critical values for the chi-square distribution with n-1 df Step 4: Compute the confidence interval for  2 based on the formula below Step 5: Obtain confidence interval for standard deviation  by taking square roots of bounds for  2

Statistical Test for  2 Null and alternative hypotheses –1-sided (upper tail): H 0 :  2   0 2 H a :  2 >  0 2 –1-sided (lower tail): H 0 :  2   0 2 H a :  2 <  0 2 –2-sided: H 0 :  2 =  0 2 H a :  2   0 2 Test Statistic Decision Rule based on chi-square distribution w/ df=n-1: –1-sided (upper tail): Reject H 0 if  obs 2 >  U 2 =   2 –1-sided (lower tail): Reject H 0 if  obs 2 <  L 2 =  1-  2 –2-sided: Reject H 0 if  obs 2  U 2 =   /2 2 (Conclude  2 >  0 2 )

Inferences Regarding 2 Population Variances Goal: Compare variances between 2 populations Parameter: (Ratio is 1 when variances are equal) Estimator: (Ratio of sample variances) Distribution of (multiple) of estimator (Normal Data): F-distribution with parameters df 1 = n 1 -1 and df 2 = n 2 -1

Properties of F-Distributions Take on positive density over the range (0,  ) Cannot take on negative values Non-symmetric (skewed right) Indexed by two degrees of freedom (df 1 (numerator df) and df 2 (denominator df)) Critical values given in Table 9, pp Parameters of F-distribution:

Critical Values of F-Distributions Notation: F a, df1, df2 is the value with upper tail area of a above it for the F-distribution with degrees’ of freedom df 1 and df 2, respectively F 1-a, df1, df2 = 1/ F a, df2, df1 (Lower tail critical values can be obtained from upper tail critical values with “reversed” degrees of freedom) Values given for various values of a, df 1, and df 2 in Table 9, pp

Test Comparing Two Population Variances Assumption: the 2 populations are normally distributed

(1-  )100% Confidence Interval for  1 2 /  2 2 Obtain ratio of sample variances s 1 2 /s 2 2 = (s 1 /s 2 ) 2 Choose , and obtain: – F L = F , n2-1, n1-1 = 1/ F , n1-1, n2-1 –F U = F , n2-1, n1-1 Compute Confidence Interval: Conclude population variances unequal if interval does not contain 1

Tests Among t > 2 Population Variances Hartley’s F max Test –Very simple to compute Test Statistic –Must have equal sample sizes (n 1 = … = n t ) –Test based on assumption of normally distributed data –Uses special table for critical values (Table 10, p. 700) Levene’s Test –More difficult to compute by hand –No assumptions regarding sample sizes/distributions –Uses F-distribution for the test –Computed automatically by software packages (SAS,SPSS, Minitab)

Hartley’s F max Test H 0 :  1 2 = … =  t 2 (homogeneous variances) H a : Population Variances are not all equal Data: s max 2 is largest sample variance, s min 2 is smallest Test Statistic: F max = s max 2 /s min 2 Rejection Region: F max  F * (Values from Table 10, p. 700, indexed by  (.05,.01), t (number of populations) and df 2 (n-1, where n is the individual sample sizes)

Levene’s Test H 0 :  1 2 = … =  t 2 (homogeneous variances) H a : Population Variances are not all equal Data: For each group, obtain the following quantities: