# Interpreting test statistic from testing $\sigma_1^2 = \sigma_2^2$

I am running through a past test paper; The question (simplied) goes as follows - the university has had a number of 10-pin bowling ball games over the years. Participants Jeremy and Jonathon want to find out who is the best player. The scores for the last 6 games were taken:

$$\text{Jeremy}= \{116, 111, 106, 145, 120, 119\} \\ \text{Jonathon} = \{112, 150, 115, 146, 156, 130\}$$

We want to test: \begin{align} H_0&: \sigma_1^2=\sigma_2^2\\ H_1&: \sigma_1 >\sigma^2 \end{align} The unbiased estimated variance for Jeremy is $$s_1^2= 183.5$$, as for Jonathon $$s_2^2 = 348.17$$.

We use the following test statistic:

$$F = \frac{s_1^2}{s_2^2}$$ where $$F_\alpha$$ is chosen so that $$P(F > F_\alpha)$$ when $$F$$ has $$v_1 = n_1-1 \implies 6-1 = 5$$ and $$v_2 = n_2-1 \implies 6-1 = 5$$.

$$F = \frac{183.5}{348.17}\\=0.53$$ We compare this to $$F_{\frac{\alpha}{2}}=F_{.05} = 3.45$$, because $$0.53 < 3.43$$ suggesting there is insufficient evidence to support who is the better player.

When calculating the p-value I used the following r-command: $$pf(0.53, 5, 5) = 0.25$$ suggesting that there is weak evidence to support the claim.

As for the test-statistics, my question is what determines which player is $$s_1^2$$ and $$s_2^2$$ because if I chose Jonathon for $$s_1^2$$ this would produce an entirely new result in terms of the P-value, suggesting that Jonathon is the better player.

Or would I be expected to test both alternatives?

• Why would testing the variance of the scores indicate who is the better player? Why not test for a significant difference between the mean scores? May 31, 2022 at 15:50
• @jbowman that was definitely the next question and I have already attempted it. Although, I guess my question is more general when working with hypotheses to test the variance when using the F-statistic May 31, 2022 at 15:56

You are testing if $$\sigma_1>\sigma_2$$. That is, you are testing if player $$1$$ has greater variance than player $$2$$. Since the observation is that player $$2$$ has greater variance than play $$1$$, you should expect the hypothesis test to come back and say that there is no evidence of player $$1$$ having greater variance than player $$2$$.

If you test if $$\sigma_1<\sigma_2$$, you might get a significant result.

If you test if $$\sigma_1 \ne\sigma_2$$, you might get a significant result.

As far as how to determine who is player $$1$$ and who is player $$2$$, you pick that depending on what you want to test. If you want to test if Jeremy has greater variance than Jonathon, you can set them as players $$1$$ and $$2$$, respectively, and have the alternative hypothesis that you have; or you can set them as players $$2$$ and $$1$$, respectively, and have $$H_a:\sigma_2<\sigma_1$$. These will give identical results.

(I agree that the variance of the scores gives limited insight into who is the better player, however.)

• Thank you for the clarification! the question explicitly states - "Test whether the variances of the scores of each player are equal", this was not added in the post, but would it be fair to say that this suggests to test both player 1 and player 2, or would using a two-tailed rejection region do better? May 31, 2022 at 16:13
• @teslajohn I interpret that to mean an a two-sided test.
– Dave
May 31, 2022 at 16:19