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

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 . 60

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

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

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.

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

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.

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