$Y_1, Y_2, ... , Y_{12}$ are i.i.d ~ N($\mu, \sigma^2)$. Let $U = \sum_{i=1}^6 (Y_i - \overline{Y}_a)^2$ and $V = \sum_{i=7}^{12} (Y_i - \overline{Y}_b)^2$, where $\overline{Y}_a = \frac{1}{6}\sum_{i=1}^{6}Y_i$ and $\overline{Y}_b = \frac{1}{6} \sum_{i=7}^{12}Y_i$.
(a) Find the distribution for each of the following two random variables. Verify your answer.
----(i) $\sum_{i=1}^{12} \left( \frac{Y_i - \mu}{\sigma} \right)^2$
----(ii) $(\frac{1}{\sqrt{12}}\sum_{i=1}^{12} \left( \frac{Y_i - \mu}{\sigma} \right))^2$
(b) Find a and b such that $P(a \lt \frac{U}{V} \lt b) = 0.9$.
(c) Find c and d such that $P(c \lt \frac{\overline{Y_a} - \overline{Y_b}}{\sqrt{U}} \lt d) = 0.95$
(d) $W=(Y_1 - Y_2)^2 + (Y_3 - Y_4)^2 +(Y_5 - Y_6)^2 + (Y_7 - Y_8)^2 + (Y_9 - Y_{10})^2 + (Y_{11} - Y_{12})^2$. Find a g such that $P(W \gt g\sigma^2)=0.9$
What I have tried so far:
For part (i) of (a), I realized $Z_i = \frac{Y_i - \mu}{\sigma}$, which makes that whole summation become $\sum_{i=1}^{12} Z_i^2$ which should ~$\chi^2(12)$
For part (ii) of (a), I tried a similar approach, but wasn't entirely sure how to deal with the squared summation. I ended up trying to set it equal to $\frac {1}{12}(\sum_{i=1}^{12} Z_i)^2$ = $\frac {1}{12}(\sum_{i=1}^{12} Z_i^2 + \sum\sum_{i \neq j} Z_iZ_j)$, which I ended up with something like $\frac{1}{\sqrt{144}}\sum_{i=1}^{144}Z_i^2$. But this doesn't seem to follow any recognizable distribution that I can tell.
At this point, I started to get stuck. I unfortunately missed the class that this material was presented in, so I'm trying to piece together the information. As far as I can tell, most of these problems should follow $\chi^2$, T, or F distributions. Any help would be greatly appreciated. Thanks in advance.
I will give some clues without working all the parts in detail.
(a) Right track on (i). For (ii) Consider $\frac{1}{12}[\sum_i Z_i^2].$
(b) Consider F-distribution.
(c) Consider t-distribution, but be careful about degrees of freedom. Note that you have three mutually independent random variables.
(d) What is the distribution of $(Y_i - Y_2)^2?$ Consider $W/\sigma^2.$