# Significance test testing whether one variance is larger than the other

I feel like I am missing a key link and this tasks is quite easy: I am given the following averages and standard deviations:

$$\bar{x}_1 = 971$$, $$s_1 = 42$$, $$n_1 = 1000$$

$$\bar{x}_2 = 983.9$$, $$s_2 = 60.18$$, $$n_2 = 10$$

For the second dataset, I have all observations.

What is an appropriate test to check whether $$\sigma_1$$ is larger than $$\sigma_2$$? Of course I stumbled upon the F-Test, but as far as I understood it, it can only handle testing whether variances are equal or not; not whether one is larger than the other. A friend of mine used Chi-Square, but I do not see how Chi-Square would be applicable here.

Thank you and best Paul

• How would you do the F-test to check if the two variances are equal? // Did you know up front that you should be checking that $\sigma_2>\sigma_1$, or did you base that on the fact that $s_2 > s_1?$ (You wrote that you are to check if $\sigma_1 > \sigma_2$, but I assume you meant the reverse.) // You might be in a part of your course where you always assume everything is normal, but the F-test can have awful performance when that assumption is violated.
– Dave
Jul 28, 2021 at 20:18

Assuming normal data, under $$H_0: \sigma_2^2/\sigma_1^2,$$ the ratio $$F =s_2^2/s_1^2 = 60.18^2/42^2 = 2.053 \sim \mathsf{F}(9,999).$$

For a one-sided test against $$H_a: \sigma_2^2/\sigma_1^2 >1,$$ one rejects if the F-statistic exceeds the the 95th percentile $$1.8892$$ of $$\mathsf{F}(9,999),$$ called the critical value. So you do (just barely) reject at the 5% level of significance. [For a two-sided test against $$H_a: \sigma_2^2/\sigma_1^2$$ the critical values would be at $$0.299$$ and $$2.126,$$ so you would not reject against the 2-sided alternative at the 5% level.]

qf(0.95,9,999)
[1] 1.889236

qf(c(.025,0.975),9,999)
[1] 0.299400 2.126391


In terms of P-values, the P-value for the 1-sided test is $$0.031 < 0.05 = 5\%$$ so you'd reject at the 5% level. [The P-value of the 2-sided test would be double the P-value of the 1-sided test.]

1 - pf(2.053, 9, 999)
[1] 0.0310929


The figure below shows the density function of $$\mathsf{F}(9,999)$$ along with the observed value of $$F$$ (solid black vertical line) and the one-sided critical value (dotted red).

Note: Traditionally, the larger sample variance is put in the numerator of the F-statistic because many printed tables of F distributions do not give lower tail probabilities. (Of course, most software programs give probabilities in both tails.)