Inferences Regarding Population Variances 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 s2

Sampling Distribution of s2 (Normal Data) Population variance (s2) is a fixed (unknown) parameter based on the population of measurements Sample variance (s2) varies from sample to sample (just as sample mean does) When Y~N(m,s), the distribution of (a multiple of) s2 is Chi-Square with n-1 degrees of freedom. (n-1)s2/s2 ~ c2 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 7, pp. 1095-1096

Chi-Square Distributions

Chi-Square Distribution Critical Values

Chi-Square Critical Values (2-Sided Tests/CIs) c2L c2U

(1-a)100% Confidence Interval for s2 (or s) Step 1: Obtain a random sample of n items from the population, and compute s2 Step 2: Choose confidence level (1-a ) Step 3: Obtain c2L and c2U from the table of critical values for the chi-square distribution with n-1 df Step 4: Compute the confidence interval for s2 based on the formula below Step 5: Obtain confidence interval for standard deviation s by taking square roots of bounds for s2

Statistical Test for s2 Null and alternative hypotheses Test Statistic 1-sided (upper tail): H0: s2  s02 Ha: s2 > s02 1-sided (lower tail): H0: s2  s02 Ha: s2 < s02 2-sided: H0: s2 = s02 Ha: s2  s02 Test Statistic Decision Rule based on chi-square distribution w/ df=n-1: 1-sided (upper tail): Reject H0 if cobs2 > cU2 = ca2 1-sided (lower tail): Reject H0 if cobs2 < cL2 = c1-a2 2-sided: Reject H0 if cobs2 < cL2 = c1-a/2 2 (Conclude s2 < s02) or if cobs2 > cU2 = ca /22 (Conclude s2 > s02 )

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 df1 = n1-1 and df2 = n2-1

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 (df1 (numerator df) and df2 (denominator df)) Critical values given in Table 8, pp 1097-1108 Parameters of F-distribution:

Critical Values of F-Distributions Notation: Fa, df1, df2 is the value with upper tail area of a above it for the F-distribution with degrees’ of freedom df1 and df2, respectively F1-a, df1, df2 = 1/ Fa, 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, df1, and df2 in Table 8, pp 1097-1108

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

(1-a)100% Confidence Interval for s12/s22 Obtain ratio of sample variances s12/s22 = (s1/s2)2 Choose a, and obtain: FL = F1-a/2, n2-1, n1-1 = 1/ Fa/2, n1-1, n2-1 FU = Fa/2, 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 Fmax Test Must have equal sample sizes (n1 = … = nt) Test based on assumption of normally distributed data Uses special table for critical values Levene’s Test No assumptions regarding sample sizes/distributions Uses F-distribution for the test Bartlett’s Test Can be used in general situations with grouped data Uses Chi-square distribution for the test

Hartley’s Fmax Test H0: s12 = … = st2 (homogeneous variances) Ha: Population Variances are not all equal Data: smax2 is largest sample variance, smin2 is smallest Test Statistic: Fmax = smax2/smin2 Rejection Region: Fmax  F* (Values from class website, indexed by a (.05, .01), t (number of populations) and df2 (n-1, where n is the individual sample sizes)

Levene’s Test H0: s12 = … = st2 (homogeneous variances) Ha: Population Variances are not all equal Data: For each group, obtain the following quantities:

Bartlett’s Test General Test that can be used in many settings with groups H0: s12 = … = st2 (homogeneous variances) Ha: Population Variances are not all equal MSE ≡ Pooled Variance