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So, this is the question I'm working on:

Suppose we observe a random sample of five measurements: 10, 13, 15, 15, 17, from a normal distribution with unknown mean $µ_1$ and unknown variance $σ_1^2$. A second random sample from another normal population with unknown mean $µ_2$ and unknown variance $σ_2^2$ yields the measurements: 13, 7, 9, 15, 11.

b. Use the pivotal method (and a pivotal statistic with F distribution) to derive a 95% confidence interval for $σ_2/σ_1$. Work it out for these data. And test the null hypothesis that $σ_2 = σ_1$ at the 5% level of significance. [6] (recall that $F_{\nu_1,\nu_2,a} = 1/F_{\nu_2,\nu_1,1-a}$).

So, I'm completely at a loss as to how I can use the pivotal method on the F distribution. Please help me.

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  • $\begingroup$ Welcome to the site. I have tidied up your question a little. Please check the faq, especially the later part of the section relating to asking questions, where it relates to homework. Please add the self-study tag and read the self-study tag wiki. With those pieces of information in mind ... what do you understand about the pivotal method and what have you tried? $\endgroup$ – Glen_b Apr 26 '13 at 3:26
  • $\begingroup$ More specifically, what is a pivotal quantity? $\endgroup$ – Glen_b Apr 26 '13 at 3:29
  • $\begingroup$ The only thing I could think of doing was using this F distribution -> F = w1/v1 / w2/v2 and I ended up getting 28/40. But, I'm not sure what this represents and I'm unsure of how to continue. And the pivotal quantity is a distribution that doesn't depend on unknown parameters .. So, does that mean that this σ2/σ1 does not depend on µ and σ2. $\endgroup$ – user24872 Apr 26 '13 at 3:35
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You're working the question from the wrong end.

Forget the F; it doesn't come into it yet. First find a nice pivotal quantity, then worry about its distribution.

Check the actual definition of a pivotal quantity here (read as far as the '[1]').

(What important piece of information did you leave out of the definition?)

Now, just think about trying to estimate $\sigma_1/\sigma_2$. How might you estimate it?

Would the distribution of that estimate depend on $\sigma_1/\sigma_2$? Does the estimator contain $\sigma_1$ or $\sigma_2$?

How might you fix both issues at the same time to turn it into a pivot? That is, what do you do to the estimator in order to make it also a function of $\sigma_1$ and $\sigma_2$, while its distribution no longer depends on them?

If you get that far, you'll get a pivot, but it won't necessarily have an F-distribution. [If that happens, the only thing left then is to understand the relationship between what you have and something that does have an F distribution.]

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  • $\begingroup$ So, I if I wanted to estimate σ1/σ2 I'd calculate the sample variances of the data in the question and find get the sample standard deviation, which would estimate σ1/σ2. $\endgroup$ – user24872 Apr 26 '13 at 4:07
  • $\begingroup$ So, the estimator depends on Y1 .. Yn and σ1/σ2, but its distribution does not depend on σ1/σ2 or any unknown parameters. But, I don't see how I'm supposed to work from here. $\endgroup$ – user24872 Apr 26 '13 at 4:12
  • $\begingroup$ The distribution of what wouldn't depend on $\sigma_1/\sigma_2$? What is your estimator, and what is the resulting pivot? $\endgroup$ – Glen_b Apr 26 '13 at 4:15
  • $\begingroup$ the estimator would be s12/s22 but I'm not sure what the resulting pivot would be $\endgroup$ – user24872 Apr 26 '13 at 4:16
  • $\begingroup$ Make sure to keep clear what you're estimating. Sample variances to estimate population variances rather than s.d.'s, for example. It doesn't matter which you work with as long as you stick to one. So fine, work with variances in both. The distribution of that estimator certainly depends on the ratio of population variances. $\endgroup$ – Glen_b Apr 26 '13 at 4:21

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