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Homework Help: Help deriving a chi-square confidence interval

  1. Oct 16, 2012 #1
    1. The problem statement, all variables and given/known data

    Assume Y1, ... , Yn is a sample of size n from a gamma distributed population with α = 2 and unknown β

    a) Use the method of moment generating functions to show that

    [itex] U = 2 \sum_1^n Y_i / β [/itex]

    is a pivotal quantity and has a chi-square distribution with 4n d.f.
    b) Use U to derive a 95% confidence interval for β
    c) If a sample of size n = 5 yields y-bar = 5.39, use the result from b to give a 95% confidence interval for β

    2. Relevant equations

    2Yi/β has a chi square with 4 d.f.

    3. The attempt at a solution

    I've done part a) easily, just take E[etU] = (1-2t)-2n so U has a chi-square distribution with 4n d.f.

    Par b is giving me more trouble, since I don't know how to start it. Usually, I would look at a table to find numbers (a,b) so that
    P(a < U < b) = 0.95
    by taking first P(U < a) = 0.025 and P(b < U) = 0.025 from the tables.

    But since I don't know the degrees of freedom, how do I know which values to choose?
  2. jcsd
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