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Assume that $X_1, \ldots, X_n$ are i.i.d. with the normal distribution that has mean $μ$ and variance $1$. Suppose that we wish to test the hypotheses
$$ H_0: μ ≤ μ_0, \\ H_1: μ > μ_0. $$ Consider the test that rejects $H_0$ if $Z≥c$, where $Z$ is defined in Eq. $(9.1.10)$: $$ Z = \frac{\bar{X}-\mu_0}{1/\sqrt{n}}. $$

a) Show that $\Pr(Z\geq c\mid\mu)$ is an increasing function of $\mu$.

b) Find $c$ to make the test have size $\alpha_0$.


This is from de Groot, Statistics, 4th edition, exercise 8, page 548.

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[ Although it's tagged as "self-study, and the question doesn't show any substantial work from the OP, I'm answering this because it's quite old and got no responses. ]

a) The power function of the proposed test procedure is \begin{align*} \pi(\mu) &= \Pr\{\text{Reject } H_0 \mid \mu\} \\ &= \Pr\{Z\geq c\mid \mu\} \\ &= \Pr\left\{ \frac{\bar{X}-\mu_0}{1/\sqrt{n}} \geq c \;\;\Bigg\vert\;\; \mu \right\} \\ &= \Pr\left\{ \frac{\bar{X}-\mu}{1/\sqrt{n}} \geq c + \frac{\mu_0-\mu}{1/\sqrt{n}} \;\;\Bigg\vert\;\; \mu \right\} \\ &= 1-\Phi\!\left(c + \frac{\mu_0-\mu}{1/\sqrt{n}}\right), \end{align*} which is an increasing function of $\mu$.

b) The test size is $\pi(\mu_0)$. Therefore, $c=\Phi^{-1}(1-\alpha_0)$.

n <- 25
mu_0 <- 1
alpha_0 <- 0.05
c <- qnorm(1-alpha_0)
power <- function(mu) 1 - pnorm((mu_0-mu)/(1/sqrt(n)) + c)
plot(power, xlim = c(0.7, 2.1))
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