Sample mean and confidence interval

[michael98764]

Hi, I am Michael.
I have a question about sample mean n confidence interval.

Let X1,X2, . . . ,Xn denote a random sample of a Gamma(3,λ) and X-bar
is the sample mean.
(a) Describe the distribution of the sample mean X-bar.
I am quite confused about sample mean of distributions. Is it the chi squared? How should I should describe?

(b) Use (a) to construct a lower 95% confidence interval for λ, of the
form (0, U)
(c) Use (a) to construct an upper 95% confidence interval for λ, of
the form (L,∞)

For (b) n (c), are we gonna use pivotal quantity method 2nλ*X-bar~Gamma (n,1/2)=χ2(2N)? If yes, how can i find χ2 for (1-α/2) and χ2 for (α/2)?


Please HELP!!!
Thanks in advance for any help!!!!