Inferences Concerning Variances Math 4030 – 10a Inferences Concerning Variances

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

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

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

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

Example 1. A sample of size 20 (from a normally distributed population) results a sample variance of 0.00012. Construct the 90% confidence interval for estimating the population standard deviation .

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.

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)

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 518-519 60

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

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.

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.)

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.)