Raising the bar

Chapter 9. Hypothesis Testing

Unit 9.6. Testing for equality of two sample variances

Definition 9.6 ($latex F$-distribution). Let $latex \chi_m^2$ and $latex \chi_n^2$ be independent. Then

$latex F_{m, n} = \frac{\chi_m^2 / m}{\chi_n^2 / n}$

is called an $latex F$-distribution with $latex (m, n)$ degrees of freedom.

Comments. (1) Equivalently, $latex F$ can be defined by

$latex F_{m, n} = \frac{\sum_{i=1}^m z_i^2 / m}{\sum_{j=1}^n z_{m+j}^2 / n}$

where $latex z_1, \ldots, z_m, z_{m+1}, \ldots, z_{m+n}$ are independent standard normal.

(2) From the definitions of $latex t$ and $latex F$ it follows that $latex t_n^2 = F_{1, n}$. This fact is used in econometrics (a $latex t$ test of significance of a regression coefficient is equivalent to the $latex F$ test of significance of that coefficient).

(3) We know that for independent samples from two normal populations

$latex \frac{(n_X - 1) s_X^2}{\sigma_X^2} = \chi_{n_X - 1}^2, \quad \frac{(n_Y - 1) s_Y^2}{\sigma_Y^2} = \chi_{n_Y - 1}^2.$

Hence,

$latex \frac{s_X^2 / \sigma_X^2}{s_Y^2 / \sigma_Y^2} = \frac{\chi_{n_X - 1}^2 / (n_X - 1)}{\chi_{n_Y - 1}^2 / (n_Y - 1)} = F_{n_X - 1, n_Y - 1}.$

(4) To test the null $latex H_0: \sigma_X^2 = \sigma_Y^2$ against the alternative $latex H_a: \sigma_X^2 \ne \sigma_Y^2$ choose $latex \alpha$, find the critical values for the two-tail test from

$latex P(F > F_{n_X - 1, n_Y - 1, 1 - \alpha/2}) = 1 - \alpha/2, \quad P(F > F_{n_X - 1, n_Y - 1, \alpha/2}) = \alpha/2.$

The area between two critical values $latex F_{n_X - 1, n_Y - 1, 1 - \alpha/2}, F_{n_X - 1, n_Y - 1, \alpha/2}$ will be $latex 1 - \alpha$, see Figure 9.6. The critical values can be found using Table 14.9 and Excel.

Figure 9.6. Cut-off value for $latex F$ distribution.

Then under the null

$latex 1 - \alpha = P(F_{n_X - 1, n_Y - 1, 1 - \alpha/2} < F < F_{n_X - 1, n_Y - 1, \alpha/2})$

$latex = P\left( F_{n_X - 1, n_Y - 1, 1 - \alpha/2} < \frac{s_X^2}{s_Y^2} < F_{n_X - 1, n_Y - 1, \alpha/2} \right).$

The null is rejected if $latex s_X^2 / s_Y^2$ either exceeds the upper critical value or is smaller than the lower critical value.

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