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In this worked-out solution, I'm convinced there is a typo:

enter image description here

In standarizing the variable, I understand how typically, we're supposed to subtract the mean from the variable in the numerator, so why are we adding (T+1/B_0)? Shouldn't it be subtraction? The rest makes sense. Thanks!

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    $\begingroup$ On a different issue I don't see how you can get a UMP test when you are using a normal approximation. $\endgroup$
    – Michael R. Chernick
    Commented Apr 16, 2017 at 19:16
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    $\begingroup$ worth pointing out the source? Since these are from 2006 course notes you might have trouble giving feedback (but it does look like the professor is still around) $\endgroup$
    – Ben Bolker
    Commented Apr 16, 2017 at 20:06
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    $\begingroup$ @MichaelChernick The argument shows that a test based on $T$ is equivalent to a likelihood ratio test; the approximation comes only in calculating the critical value (i.e. the test is still most powerful, what's approximate is the significance level -- instead of a 5% test you get roughly 4.4% by using the normal approximation). Since $-T$ is itself gamma-distributed, one can compute an exact rejection rule -- "reject if $T\geq -0.8414/\beta_0$" (instead of the normal approximation's "reject if $T\geq -.8355/\beta_0$") but it's essentially beside the point to the main issue. $\endgroup$
    – Glen_b
    Commented Apr 20, 2017 at 7:51
  • $\begingroup$ Okay Glen I see your point. But the choice of the critical value is not exactly what would be used for the UMP test. $\endgroup$
    – Michael R. Chernick
    Commented Apr 20, 2017 at 7:57

1 Answer 1

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We can see from the information there that $E(X_i)=1/\beta_0$ (under the null, whence $E(\bar{X}_n) = 1/\beta_0$), and that $T=-\bar{X}_n$.

Consequently the expected value of $T$ is $-1/\beta_0$ and so $T-E(T)=T+1/\beta_0$.

There's no typo, it's doing the right thing there.

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