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404    8. Tests of Hypotheses

                                 α. See the Examples 8.3.6-8.3.7.

                                      A final form of the MP or UMP level a test is written down in the
                                     simplest implementable form. That is, having fixed α ∈ (0, 1), the
                                                                               c
                                      cut-off point of the test defining the regions R, R  must explicitly
                                          be found analytically or from a standard statistical table.
                                    Remark 8.3.3 We note that, for all practical purposes, the MP level α test
                                 given by the Neyman-Pearson Lemma is unique. In other words, if one finds
                                 another MP level α test with its test function ψ* by some other method, then
                                 for all practical purposes, ψ* and ψ will coincide on the two sets {X ∈ χ  :
                                                                                                n
                                 L(x; θ ) > kL(x; θ )} and {x ∈ χ  : L(x; θ ) < kL(x; θ )}.
                                                             n
                                      1         0                    1        0
                                       Convention: k is used as a generic and nonstochastic constant.
                                            k may not remain same from one step to another.
                                    Example 8.3.1 Let X , ..., X  be iid N(µ, σ ) with unknown µ ∈ ℜ, but
                                                                         2
                                                            n
                                                      1
                                 assume that σ ∈ ℜ +  is known. With preassigned α ∈ (0, 1) we wish to
                                 derive the MP level α test for H  : µ = µ  versus H  : µ = µ  where µ  > µ  and
                                                            0     0        1      1       1   0
                                 µ , µ  are two known real numbers. Both H , H  are simple hypotheses and
                                     1
                                  0
                                                                      0
                                                                          1
                                 the Neyman-Pearson Lemma applies. The likelihood function is given by













                                        Figure 8.3.1. (a) Standard Normal PDF: Upper 100a% Point
                                      (b) Probability on the right of z  Is Larger Under H  (darker plus
                                                                                 1
                                                                 a
                                         lighter shaded areas) than Under H  (lighter shaded area)
                                                                       0
                                 The MP test will have the following form:

                                 that is, we will reject the null hypothesis H  if and only if
                                                                     0
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