Question

Upper level statistics question based on confidence intervals

Answer 1

So my question comes mainly from part c) and d), i got for part a) that because the mean and variance are both unknown, the confidence interval should look like this\[100(1-a)=P(\frac{ (n_x-1)S _{x}^{2} }{ \chi^{2}_{a/2,n-1}}\le \sigma^2 \le \frac{ (n_x-1)S _{x}^{2} }{ \chi^{2}_{1-a/2,n-1}})\]and i believe similarly that part b) will be the same answer just with the x's replaced by y's as it would correspond to that sample instead, my only issue for b) is that i'm unsure if it's implied that both mean and vriance are unknown again or if they are known.

Answer 2

and since that's my two-sided interval, and i assume that X and Y have the same distribution where both means and variances are unknown, i write that the answer to a) should be\[[\frac{ (n_x-1)S _{x}^{2} }{ \chi _{a/2,n-1}^{2} },\infty)\]

Answer 3

and then for part b), the lower confidence interval should be\[[0,\frac{ (n_y-1)S _{y}^{2} }{ \chi^{2}_{1-a/2,n-1}})\]

Answer 4

your answers in a) and b) look right to me!

As for c):
Recall the \(F\) distribution being a ratio of 2 chi-square distributions. In this case:

\[\large \frac{(n_x-1)S_x^2}{\sigma^2_x}\sim \chi^2_{n_x-1}\]\[\large \frac{(n_y-1)S_y^2}{\sigma^2_y}\sim \chi^2_{n_y-1}\]
So, \[\Large F= \frac{ \frac{(n_y-1)S_y^2}{\sigma^2_y}/(n_y-1)}{\frac{(n_x-1)S_x^2}{\sigma^2_x}/(n_x-1)}=\frac{\sigma^2_xS^2_y}{\sigma_y^2S_x^2} \sim F(n_y-1,n_x-1)\]
Since the F distribution is free of parameters, we can use it as a pivot to find a confidence interval for \(\sigma^2_x/\sigma^2_y\).

d)
Then you have that \[\large\begin{align} P\left(F_{1-\alpha/2}(n_y-1,n_x-1) <F<F_{\alpha/2}(n_y-1,n_x-1)\right)&=1-\alpha\\
P\left(F_{1-\alpha/2}(n_y-1,n_x-1) <\frac{\sigma_x^2S_y^2}{\sigma^2_yS_x^2}<F_{\alpha/2}(n_y-1,n_x-1)\right)&=1-\alpha\\
\end{align}\]
...
\[ P\left(F_{1-\alpha/2}(n_y-1,n_x-1)\frac{S_x^2}{S_y^2} <\frac{\sigma^2_x}{\sigma_y^2}<F_{\alpha/2}(n_y-1,n_x-1)\frac{S_x^2}{S_y^2} \right)=1-\alpha\]

And so the confidence interval should be obvious from this.

Answer 5

free of unknown parameters*

Answer 6

Thank you so much, as always you've helped me understand something that i couldn't figure out on my own! I really need to start memorizing the important distribution relations, like as you pointed out, an F distribution being the ratio of 2 chi-squared distributions, it's just a lot to remember sometimes.

Answer 7

np :)! Yeah it can be overwhelming to remember all those distributions.. especially when you first see them. But it's well worth remembering them as you often need them, and if you take other stats courses, you will most likely use them too there.