12-49. Hypothesis Tests for the Coefficient of Variation Under Random Sampling.

We may want to determine if a set of sample random variables

{X1, . . . ,Xn} could have been drawn from a population having a given coefficient of variation V0 = (σ/μ)0. For instance, suppose that in 1990 a large cross-section of countries was known to have a coefficient of variation of per capita income levels equal to V1990. Has the coefficient of variation calculated from a random sample of these countries for the year 2000 changed significantly from V1990? If, for instance, there has been a statistically significant decline in the value of this statistic over time, then we have some sample evidence that convergence of per capita incomes has occurred; that is, the countries have become more similar or uniform with respect to per capita income levels.

Suppose we want to test H0 : V = V0, against the two-tail alternative H1 : V
= V0 at the α = P(TIE) level of significance. If the sampled population is N(μ, σ) with V = σ/μ ≤ 0.66, then, for a sample size n ≥ 10, the test statistic ZV =

√

n − 1 S/X − Vo

Vo!V2 o

+ 0.5 is approximately N(0, 1). Then the critical region R = {zV/|zV| ≥ zα/2}, where zV is the sample realization of ZV. For one-tailed alternatives

(involving either H0 : V ≤ V0 vs. H1 : V ≤ V0 or H0 : V ≥ V0 versus H1 : V < V0), this test works well for n ≥ 11 when sampling from a normal population with V ≤ 0.33. In this instance R = {zV|zV| ≥ zα} for H1 : V > V0 or R = {zV |zV| ≤ −zα} when H1: V < V0.

A sample of size n = 25 is taken from a N(μ, σ) population with the result that ¯x = 1.52, s = 4.67. For α = 0.05, test H0 : V = V0 = 0.35 against H1 : V < 0.35.