On the basis of data provided by a Romac salary survey, the variance in annual salaries for seniors in public accounting firms is approximately 2.1 and the variance in annual salaries for managers in public accounting firms is approximately 11.1. The salary data were provided in thousands of dollars. Assuming that the salary data were based on samples of 25 seniors and 26 managers, test the hypothesis that the population variances in the salaries are equal. At $\alpha = .05$ level of significance, what is your conclusion?

Solution

Let $X$ denote the voltage output for brand A batteries and $Y$ denote the voltage output for brand B batteries.

Given data is as follows :

. Brand A Brand B
Sample Size $n_1= 25$ $n_2=26$
Sample variance $s_1^2 = 11.1$ $s_2^2 = 2.1$

Step 1 Hypothesis testing problem

The hypothesis testing problem is

$H_0 : \sigma^2_1 = \sigma^2_2$ against $H_1 : \sigma^2_1 \neq\sigma^2_2$ ($\textit{two-tailed}$)

Step 2 Test Statistic

The test statistic for testing above hypothesis testing problem is

$$ \begin{aligned} F=\frac{s_1^2}{s_2^2} \end{aligned} $$
The test statistic $F$ follows $F$ distribution with $n_1-1= 25-1 = 24$ and $n_2-1=26-1= 25$ degrees of freedom.

Step 3 Level of significance

The significance level is $\alpha = 0.05$.

Step 4 Critical value(s)

As the alternative hypothesis is $\textit{two-tailed}$, the critical value of $F$ for $24$ and $25$ degrees of
freedom and $\alpha = 0.05$ level of significance $\text{are}$ $\text{0.443 and 2.242}$.

F-test-critical-region
F-test-critical-region

The rejection region (i.e. critical region) is $\text{F < 0.443 or F > 2.242}$.

Step 5 Computation

The test statistic under the null hypothesis is
$$
\begin{aligned}
F_{obs} &=\frac{s_1^2}{s_2^2}\
&= \frac{11.1}{2.1}\
&= 5.2857
\end{aligned}
$$

Step 6 Decision (Traditional approach)

The test statistic is $F_{obs} =5.2857$ which falls $\textit{inside}$ the critical region, we $\textit{reject}$ the null hypothesis.

OR

Step 6 Decision ($p$-value approach)

This is a $\textit{two-tailed}$ test, so the p-value is the area to the left of the test statistic ($F_{obs}=5.2857$) is p-value = $0.0001$.

The p-value is $0.0001$ which is $\textit{less than}$ the significance level of $\alpha = 0.05$, we $\textit{reject}$ the null hypothesis.

Interpretation

We conclude that at $\alpha=0.05$ level of significance the population variances in the salaries for managers in public accounting firms and salaries for seniors in public accounting firms are not equal.