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The Statement of the Problem:

Let $X_1,..., X_n$ be a random sample from $N(\mu, 1)$. Consider testing $$ H_0: \mu=0 \quad \text{vs.} \quad H_a: \mu=1. $$

Let the rejection region be $C = \{ \overline X > c\}$
(a) Find $c$ so that the test has size $\alpha = 0.01$.
(b) Find the power under $H_a$, i.e. find $\gamma_C(1).$
(c) Show that $\gamma_c(1) \to 1$ as $n \to \infty$. How do you interpret this?

Where I Am:

I apologize if this is really basic, but I can't figure out how to do this. I know how to test for alternative hypotheses that are state that the mean is not a value, or larger than or smaller than some value; and I know how to test for the difference between means of different samples, but this doesn't really make sense to me. If someone could shed a little light on this, I'd appreciate it.

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  • $\begingroup$ is it a homework? $\endgroup$
    – Simone
    Commented Jul 30, 2015 at 2:14
  • $\begingroup$ Yes, it is. (Here's some more characters). $\endgroup$ Commented Jul 30, 2015 at 2:15
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    $\begingroup$ Testing an alternative of this form is essentially the same as a composite alternative. The question is already specifying the form of the rejection region, so you want to consider the distribution of $\bar{X}$ when $\mu = 0$ and when $\mu = 1$, and how often you will be rejecting $H_0$ for different values of $c$ in either situation. $\endgroup$
    – dsaxton
    Commented Jul 30, 2015 at 2:16
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    $\begingroup$ @Simone, it is best to ask the OP to add the SS tag & read it's wiki, rather than add the tag for them. That way, there is a better chance they will be familiar w/ our policy. If they don't add the tag, you can vote to close. $\endgroup$ Commented Jul 30, 2015 at 3:05
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    $\begingroup$ @gung Will do! Sorry. First-time poster. $\endgroup$ Commented Jul 30, 2015 at 3:06

3 Answers 3

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$\bar{X}$, the sample mean, is distributed as $N(0,n^{-1})$ when $H_0$ is true and as $N(1,n^{-1})$ when $H_1$ is true. (Here, the parameters are $\mu$ and $\sigma^2$

Thus, we want $c$ to be that number for which $P\{\bar{X} > c \mid H_0\}$ equals $0.01$. But, $$P\{\bar{X} > c \mid H_0\} =Q\left(c\sqrt{n}\right)$$ where $Q(\cdot)$ is the complementary standard normal distribution function, and a quick look at tables (or resort to an R-gument) shows that $Q(2.33) \approx 0.01$. Hence,

$$c \approx \frac{2.33}{\sqrt{n}}\tag{1}$$

It follows that

$$P\{\bar{X} > c \mid H_1\} = Q\left((c-1)\sqrt{n}\right) \approx Q\left(2.33 - \sqrt{n}\right)\tag{2}$$

Since the argument of $Q$ in $(2)$ diverges to $-\infty$ as $n \to \infty$ (and so the value of $Q(\cdot)$ approaches $1$), we have part (c) as well. The interpretation is that while the threshold $c$ converges to $0$ as $n\to\infty$ (cf. $(1)$), the variance of $\bar{X}$ is converging to $0$, and thus, when $H_1$ is true, the probability that $\bar{X} \sim N(1,\sqrt{n^{-1}})$ exceeds the threshold $c$ is close to $1$. Even though that threshold is quite close to the mean $1$ in absolute terms, it is zillions of standard deviations away from the mean $1$.

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Hints:

(a) Under $H_0$ we have $\bar{X} \sim$ normal$(0, 1 / n)$ and so $\bar{X} \sqrt{n} \sim$ normal$(0, 1)$, and so we want to find a $c$ such that $P(\bar{X} \sqrt{n} > c \sqrt{n}) = \alpha$. It's perhaps a bit confusing that the author framed the question in this way since here $c$ will depend on $n$ in order to get proper size.

(b) You actually can't answer this without knowing $n$, because the power depends on $n$. I'm beginning to wonder if the author made a mistake.

(c) Suppose we've chosen some $z$ to use as the lower bound of a rejection region of the form $\{ \bar{X} \sqrt{n} > z \}$ (i.e., $z = c \sqrt{n}$). When $H_1$ is true we can look at the event of rejection in this way $\{ \bar{X} \sqrt{n} > z \} = \{ (\bar{X} - 1)\sqrt{n} + \sqrt{n} > z \} = \{ (\bar{X} - 1)\sqrt{n} > z - \sqrt{n} \}$. What happens to the probability of this event as $n \to \infty$?

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For (b), you will reject $H_0$ if $\frac{\bar{X}-0}{\sigma/\sqrt{n}}\geq Z_{\alpha}$ (where $\sigma=\sqrt{1/n}$ in fact, it should be sample variance)

The power is the probability to reject $H_0$ if the $H_1$ is true

$\gamma_{(c)}(1)=P_{\mu}( \bar{X} \geq Z_{\alpha}\sigma/\sqrt{n} )=P_{\mu}(\frac{\bar{X}-1}{\sigma/ \sqrt{n}} \geq \frac{Z_{\alpha}\sigma/\sqrt{n}-1}{\sigma/\sqrt{n}})=P_{\mu}(\frac{\bar{X}-1}{\sigma/ \sqrt{n}}\geq Z_{\alpha}-\frac{1}{\sigma/\sqrt{n}})=1-\Phi(Z_{\alpha}-n)$

($\sigma=\sqrt{1/n}, \Phi $ is CDF of standard normal distribution)

So your power function is $1-\Phi(Z_{\alpha}-n)$. It depends on both $\alpha $ and $n$.

When $\alpha=0.01$ The power is $1-\Phi(1.64485-n)$

You also can see when $n\rightarrow\infty, \Phi(1.64485-n)\rightarrow0$ so the power will be 1

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