Question

Two random samples denoted by X1 and X2 of sizes 10 and 20 respectively have unbiased sample variances 0.0003 and 0.0001 respectively. Assuming that the populations from which the samples have been drawn are normal, determine whether the variance of the first sample is significantly different from the variance of the second sample.

Answer 1

Let the following notations represent population 1 and 2.

Population1

$n_1=10$

$Sample\displaystyle\space variance=S^2_1=0.0003$

Population 2

$n_2=20$

$Sample\displaystyle\space variance=S^2_2=0.0001$

Hypothesis tested is,

$H_0: \sigma^2_1=\sigma^2_2\displaystyle\space against\displaystyle\space H_1:\sigma^2_1\not=\sigma^2_2$

To test this hypothesis, F distribution is used to establish a decision criteria as follows.

The F test statistic is given as,

$F_c=S^2_1/S^2_2=0.0003/0.0001=3$

$F_c=3$ is compared with the $F$ table with $(n_1-1)=10-1=9$ and $(n_2-1)=20-1=19$ degrees of freedom at $\alpha=0.05$

This table value $F_{\alpha/2,9,19}=F_{0.05/2,9,19}=F_{0.025,9,19}=2.88$. $\alpha/2$ is used because it is a two-sided test.

Since $F_c=3$ is greater than the table value $F_{0.025,9,19}=2.88$, we reject the null hypothesis and conclude that there is sufficient evidence to show that the variance of the first sample is significantly different from the variance of the second sample.