3.6 Two population variances

• Testing hypotheses about the equality of two population variances requires yet another continuous distribution: F-distribution

• One of the unique features of the $F-$distribution, unlike $t-$distribution or $\chi^2-$distribution, is that it’s characterized by two types of degrees of freedom, known as numerator degrees of freedom $df_1$ and denominator degrees of freedom $df_2$

• The degrees of freedom are called numerator and denominator because an $F$ random variable is actually the ratio of two $\chi^2$ random variables, each of which has its own number of degrees of freedom

• In this case, the null hypothesis is written as follows:

$$\begin{equation} H_0:~~\sigma_1^2=\sigma_2^2 \tag{3.18} \end{equation}$$

• The null hypothesis is that the two population variances are equal. This is not rejected unless strong evidence indicates otherwise

• The alternative hypothesis can take one of three forms:

$$\begin{align} H_1:&~~\sigma_1^2 \ne \sigma_2^2 &\text{two-tailed test} \\ \\ H_1:&~~\sigma_1^2 < \sigma_2^2 &\text{left-tailed test} \\ \\ H_1:&~~\sigma_1^2 > \sigma_2^2 &\text{right-tailed test} \\ \tag{3.19} \end{align}$$

• The appropriate test statistic follows the $F-$distribution

$$\begin{align} F&=\frac{S_1^2}{S_2^2} \sim F(df_1,~df_2) \\ \tag{3.20} \end{align}$$

• Here is what each term means:

$$\begin{align} S_1^2&~\text{is the varaince of the first sample (chosen from population 1)} \\ \\ S_2^2&\text{is the variance of the second sample (chosen from population 2)} \\ \\ df_1&=(n_1-1)~\text{is the numerator degrees of freedom} \\ \\ df_2&=(n_2-1)~\text{is the denominator degrees of freedom} \\ \\ n_1&~\text{is the size of the first sample} \\ \\ n_2&~\text{is the size of the second sample} \\ \end{align}$$

Example 3.11 An investor wants to determine whether two portfolios have the same volatility. He takes a sample of ten stocks from each portfolio. The sample standard deviation of the first portfolio is 26 percent, and the sample standard deviation of the second portfolio is 24 percent. Compute the p-value in Excel using function =F.DIST.RT(f;df1;df2;TRUE).

Example 3.12 At $1\%$ significanve level test the hypothesis that the variation in advertising costs of the listed companies is greater than variation in advertising costs of the companies not listed on the stock exchange. Use the data from Excel file. Obtain the results by Data Analysis ToolPak.