1. Math 4030 – 10b Inferences Concerning Variances: Hypothesis Testing
Estimation
Hypothesis Testing

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 . 60

4. How to find the chi-square cut-off scores?
Integrals: solve such that
and

6. Confidence Interval for Population Variance:
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 with degree of freedom n – 1, find and
Step 2. The (1-)100% confidence interval for
the population variance is

7. Hypothesis Testing regarding Variance or standard deviation
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. 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; 60

10. Hypothesis Testing to compare two Variances
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.

11. Property of F distribution:
By definition, F is the cut-off value such that
Case 1.
compare with
Case 2.
compare with
Case 3.
compare with