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Math Help - What goes wrong?

  1. #1
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    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
    Last edited by noob mathematician; October 17th 2009 at 04:55 AM.
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  2. #2
    Moo
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    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,s), then aM~N(am,as)

    Can you try to do it ?

    Also, please post all the information you're given, because I feel like there's not everything here
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  3. #3
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    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)?
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