1. ## What goes wrong?

$Y_1, Y_2$ are independent IID samples from two normal populations sharing same variance

I can't show that the distribution of $(\frac{\overline{Y_1}-\overline{Y_2}}{S\sqrt{1/n_1+1/n_2}})=t_{(n_1+n_2-2)}$

What I did was to make use of the definition: $t_n\sim\frac{Z}{\sqrt{\chi ^2_n/n}}$

My Z is $\frac{(\overline{Y_1}-\overline{Y_2})-0}{\sigma\sqrt{1/n_1+1/n_2}}$

but I can't continue after this

2. Hello,

But Z has to be a normal distribution N(0,1)

Assuming Y1 and Y2 are iid samples of N(0,1), then we have :
- Y1-Y2 follows a normal distribution (because they're independent)
- E(Y1-Y2)=0
- Var(Y1-Y2)=n1+n2

So in order to get a standard normal distribution from Y1-Y2, find a such that a(Y1-Y2) follows N(0,1)

And for that, recall that if M~N(m,s²), then aM~N(am,a²s²)

Can you try to do it ?

Also, please post all the information you're given, because I feel like there's not everything here

3. Hi, this is actually not a question but I'm trying to find out the derivation: The notes go like this:

(Two Normal Populations with Equal Variance)

Let $Y_{11},....,Y_{1n_1}$ and $Y_{21},...,Y_{2n_2}$ be independent IID samples from two populations with means $\mu_1, \mu_2$.

Under the null hypothesis $\mu_1=\mu_2$

then the distribution $(\frac{\overline{Y_1}-\overline{Y_2}}{S\sqrt{1/n_1+1/n_2}})=t_{(n_1+n_2-2)}$

So I suppose my Z is standardised already? since $E(\overline{Y_1}-\overline{Y_2})=0$ and $Var(\overline{Y_1}-\overline{Y_2})=\sigma^2(1/n_1+1/n_2)$?