Confidence Intervals

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Main Question or Discussion Point

I've been working on this problem for a while, but I am really not sure that I'm doing it right. Here is the statement:

Let Y1; Y2; : : : ; Yn be i.i.d. from a gamma distribution with known shape parameter alpha and unknownscale parameter beta. Find a (1-alpha )% condfidence interval for the parameter . (Hint: the Minimum Variance Unbiased Estimator for beta is beta hat = Ybar/alpha )

I've attached the work that I've done, but I haven't used the MVUE, so I feel like I made a mistake in my distribution somewhere. Can anyone help me out?
hw.jpg
 

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