1. Inferences Regarding Population Variances
Chapter 7
Inferences Regarding Population Variances

2. 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

3. 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 8, pp

4. Chi-Square Distributions

5. Chi-Square Distribution Critical Values

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

7. (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 for below
Step 5: Obtain confidence interval for standard deviation s by taking square roots of bounds for s2

8. 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 )

9. 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

10. 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 9, pp
Parameters of F-distribution:

12. 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 9, pp

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

15. (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

16. Tests Among t > 2 Population Variances
Hartley’s Fmax Test
Very simple to compute Test Statistic
Must have equal sample sizes (n1 = … = nt)
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)

17. 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 Table 10, p. 700, indexed by a (.05, .01), t (number of populations) and df2 (n-1, where n is the individual sample sizes)

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