1. Inferences Concerning Variances
Math 4030 – 10a
Inferences Concerning Variances

2. Sample variance is defined as
Unbiasedness
If S2 is the variance of a random sample of size n taken from a normally distributed population with variance σ2, then
has chi-square distribution with parameter (df) ν = n – 1. 48

3. Chi-square distribution
Is a special case of Gamma distribution when
Density function:
Mean is .
Table 5 on Page 517:
Table values are cut-off scores. (Same as t-Table) 60

4. For sample of size n = 10,
95% Confidence Interval for variance?

5. Confidence Interval for Population Variance (Sec. 9.2):
Objective: Estimate the population variance.
Assumption: The population is normally distributed.
Given: Sample variance s2 from a random sample of size n.  value.
Step 1. From the chi-square table (Table 5)
with degree of freedom n – 1, find
and
Step 2. The (1-)100% confidence interval for
the population variance is

6. Example 1.
A sample of size 20 (from a normally distributed population) results a sample variance of Construct the 90% confidence interval for estimating the population standard deviation .

7. Hypothesis Testing regarding Variance or standard deviation (Sec. 9.2)
Null and alternative hypotheses regarding the population variance or standard deviation
Level of significance, tail(s) of the test.
Under the normality assumption, use Chi-square distribution to find the critical value or (for one-tail test), and (for two-tails test). And determine the critical region.
Calculate the test statistic
Make conclusion.

8. Example 2.
Playing 10 rounds of golf on his home course, a golf professional averaged 71.3 with a standard deviation of 1.32.
Test the claim that he is actually less consistent than  = 1.20 which is indicated in his profile. (Use  = 0.05)

9. Compare variances from two samples (Sec. 9.4)
If S21 and S22 are the variances of two independent random samples of sizes n1 and n2, respectively, taken from two normally distributed populations with the same σ2, then
Has F distribution with parameters
Parameters v1 and v2, called numerator and denominator degrees of freedom;
Take only positive values;
Skewed to the right;
Table 6 on Page 60

11. Limitation of the Tables: How can we find F1-?
By definition, F is the cut-off value such that
Case 1.
compare with
Case 2.
compare with
Case 3.
compare with

12. Hypothesis Testing to compare two Variances (Sec. 9.3)
Null and alternative hypotheses regarding the ratio of two population variances.
Level of significance, tail(s) of the test.
Under the normality assumption, use F distribution to find the critical value(s). And determine the critical region.
Calculate the test statistic
Make conclusion.

13. Example 3. (One Side F-test)
It is desired to determine whether there is less variability in the silver plating done by Company 1 than in that done by Company 2. If a sample of size 10 from Company 1’s work and a sample of 16 from Company 2’s work yield
Test the null hypothesis 1 = 2 against the alternative hypothesis 1 < 2 at  = 0.05 level. (Assuming both populations are normally distributed.)

14. Example 4. (Two Side F-test)
It is desired to determine whether there is any difference in the variability in the silver plating done by Companies 1 and 2. If a sample of size 10 from Company 1’s work and a sample of 16 from Company 2’s work yield
Test the null hypothesis 1 = 2 against the alternative hypothesis 1 ≠ 2 at  = 0.02 level. (Assuming both populations are normally distributed.)