Derive an expression for the p-value using a test with test statistic T = n ( X ¯ n − θ 0 ) / σ

Question

Derive an expression for the   p-value using a test with test statistic   T  =      n    (                    X        ¯              n    −      θ    0    )      /    σ

Kailee Abbott · Accepted Answer

You have one sided hypothesis and monotone likelihood ratio w.r.t to the minimial sufficient statistic                     X        ¯              n  , then you can construct the most powerful test for a given test size   α using the likelihood ratio (Neyman-Pearson test). Then, using Karlin-Rubin theorem to deduce that this test is also the UMP test for every pair of   (      θ    1    ,      θ    0    ) for a given test size   α. So, let us build a MP test for       H    0    :  θ  =      θ    0  ,       H    0    :  θ  =      θ    1   ,       θ    1    &amp;gt;      θ    0  ,  L  R  =                    L            (              θ        1            )                      L            (              θ        0            )        ∝  exp  ⁡            {                                                                  X                ¯                                      n                    (                      θ            1                    −                      θ            0                    )                          σ          2                    }        ≥  k  ,hence the UMPT is  ψ  (  X  )  =  I  {                    X        ¯              n    ≥      k    ∗    }  .Thus, as under       H    0  ,                     X        ¯              n    ∼      N    (      θ    0    ,      σ    2        /    n  ) the p.value is  p  .  v  a  l  u  e  =            P                      θ        0              (                    X        ¯              n    ≥                    x        ¯              n    )  =  1  −  Φ      (                            n                (                                            x              ¯                                n                −                  θ          0                )            σ        )    .

Derive an expression for the p-value using a test with test statistic T =sqrt(n)(bar(X)_n - theta_0)/sigma

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