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

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

Last edited: