Question
Compare the power curves obtained in Problems 8.60 and 8.61 . Explain why the test in Problem 8.61 is uniformly most powerful, while the test in Problem 8.60 is not.
Step 1
60 and 8.61. Identify the key characteristics of each power curve, such as the shape, the range of parameter values, and the corresponding power values at different significance levels. Show more…
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Consider the one-sided $t$ -test for $H_{0}: \mu=\mu_{0}$ versus $H_{A 1}: \mu>\mu_{0}$ constructed in Example 4.5.4 and the two-sided $t$ -test for $t$ -test for $H_{0}: \mu=\mu_{0}$ versus $H_{1}: \mu \neq \mu_{0}$ given in $(4.6 .9) .$ Assume that both tests are of size $\alpha .$ Show that for $\mu>\mu_{0}$, the power function of the one-sided test is larger than the power function of the two-sided test.
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