Suppose that

is normally distributed with mean 0 and unknown variance

. Then

has a

with 1 df. Use the pivotal quantity

to find:

**a)** 95% confidence interval for

**b)** 95% upper confidence limit of

Solution:

I based my work on

http://www.mathhelpforum.com/math-he...-interval.html which gave me

Mr F says: All corrections are given in red:
then

Mr F says: No. The random variable here is **not** Y. It's , say.
if I make the substitution

and

then I get:

Mr F says: The substitution is a change of variable only in the integral, **NOT a change of random variable**! That 2X you had should still be a U. Also, if you introduce a change of variable, you have to change the integral terminals too! Actually, my next comment (below) shows why there's actually no point in making a substitution.
now this looks a lot like a Gamma function, but my upper bound is

and not

, which is throwing me off.

Mr F says: This equation can only be readily solved for by using technology. In fact, I'm not sure it can be solved at all without using technology. Everything I've said above applies below too.
same substitution as before

again this is looking a lot like a Gamma function, but this this my lower bound is not 0 it's

the answer in the back of the book is

**b)**
same substitution and same argument as before. I get:

with the solution in the back of the book being