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

Question

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?

Answer 1

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.)