Hi I'm working on a question that asks me to find the constant c so that
c(Xbar-X(n+1))/S has a t distribution. We're given that X1, X2, ......Xn, X(n+1) is a random sample of size n+1 from a normal distrubution.
I found that Xbar has a normal(mu,sigma^2/n) distribution, and that X(n+1) has a N(mu,sigma^2) distribution.
So then Xbar-X(n+1) has a N(0,sigma^2 (n+1)/n) distribution. I then divided the above distribution by (n-1)S^2/sigma^2 to get my t distribution.
then c=sqrt(n\n+1), which is wrong, according to the Book. They got sqrt(n-1/n+1).
The second part wants a confidence interval, and I think I need the correct c on the first part to even start it out.
any help will be appreciated
thanks

does this site have latex?\sigma^2

So then Xbar-X(n+1) has a N(0,sigma^2 (n+1)/n) distribution.


Hi tidus,

I don't think Xbar and X_{n+1} are independent. This could explain the difference.

Yes,
I guess I assumed they were independent. I suppose I need to find their joint pdf and go from there. I do not like that gnarly formula, but I will try to work it out.
Thanks

You don't need to go there. Since Xi's are independent, Cov[Xbar, X_{n+1}]=Cov[(X1+...+X_{n+1})/(n+1), X_{n+1}], you can then drop all terms from X1 to Xn...

Hi,
I went to talk to my prof after i turned this in and it turns out I was RIGHT and the book's answer was wrong. I worked on this problem for hours. I'm glad that I got it right.