/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 18 Suppose \(X_{1}, \ldots, X_{n}\)... [FREE SOLUTION] | 91影视

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Chapter 8: Problem 18

Suppose \(X_{1}, \ldots, X_{n}\) is a random sample on \(X\) which has a \(N\left(\mu, \sigma_{0}^{2}\right)\) distribution, where \(\sigma_{0}^{2}\) is known. Consider the two-sided hypotheses $$ H_{0}: \mu=0 \text { versus } H_{1}: \mu \neq 0 $$ Show that the test based on the critical region \(C=\left\\{|\bar{X}|>\sqrt{\sigma_{0}^{2} / n} z_{\alpha / 2}\right\\}\) is an unbiased level \(\alpha\) test.

Short Answer

Expert verified
The test with critical region \(C = \left\{Z : |Z| > z_{\alpha / 2}\right\}\) is proved to be an unbiased level \(\alpha\) test, since the size of the critical region is equal to \(\alpha\), the level of the test.

Step by step solution

01

Understanding the problem

We are given a sample, \(X_{1}, \ldots, X_{n}\), from a normal distribution \(N\left(\mu, \sigma_{0}^{2}\right)\) and we are testing the null hypothesis that the population mean \(\mu = 0\) against an alternative hypothesis \(\mu \neq 0\). An unbiased level \(\alpha\) test is one that rejects the null hypothesis with probability \(\alpha\) when the null hypothesis is true. We are asked to show that the test with critical region \(C=\left\{|\bar{X}|>\sqrt{\sigma_{0}^{2} / n} z_{\alpha / 2}\right\}\) is such an unbiased level \(\alpha\) test.
02

Formulate the test statistic

The test statistic is \(Z = \frac{\bar{X}}{\sigma_0/\sqrt{n}}\). According to Central Limit Theorem, \(Z\) will have \(N(0,1)\) distribution if the null hypothesis is true (\(\mu = 0\)).
03

Define critical region

The critical region is defined as \(C = \left\{Z : |Z| > z_{\alpha / 2}\right\}\), which is the set of all possible values of the test statistic for which we would reject the null hypothesis.
04

Determine the probability of rejecting the null hypothesis

We reject the null hypothesis if the calculated test statistic \(Z\) is in the critical region \(C\). The type I error rate, ie., the probability of rejecting the null hypothesis when it is true, is equal to the size of the critical region, which is \(2(1-\Phi(z_{\alpha / 2})) = \alpha\). Here, \(\Phi\) is the cumulative distribution function of the standard normal distribution. As a result, the test is an unbiased level \(\alpha\) test.

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Most popular questions from this chapter

Let \(X\) have the pdf \(f(x ; \theta)=\theta^{x}(1-\theta)^{1-x}, x=0,1\), zero elsewhere. We test \(H_{0}: \theta=\frac{1}{2}\) against \(H_{1}: \theta<\frac{1}{2}\) by taking a random sample \(X_{1}, X_{2}, \ldots, X_{5}\) of size \(n=5\) and rejecting \(H_{0}\) if \(Y=\sum_{1}^{n} X_{i}\) is observed to be less than or equal to a constant \(c\). (a) Show that this is a uniformly most powerful test. (b) Find the significance level when \(c=1\). (c) Find the significance level when \(c=0\). (d) By using a randomized test, as discussed in Example 4.6.4, modify the tests given in parts (b) and (c) to find a test with significance level \(\alpha=\frac{2}{32}\).

Let \(X_{1}, X_{2}, \ldots, X_{10}\) be a random sample from a distribution that is \(N\left(\theta_{1}, \theta_{2}\right)\). Find a best test of the simple hypothesis \(H_{0}: \theta_{1}=\theta_{1}^{\prime}=0, \theta_{2}=\theta_{2}^{\prime}=1\) against the alternative simple hypothesis \(H_{1}: \theta_{1}=\theta_{1}^{\prime \prime}=1, \theta_{2}=\theta_{2}^{\prime \prime}=4\)

Let \(X\) have a Poisson distribution with mean \(\theta\). Find the sequential probability ratio test for testing \(H_{0}: \theta=0.02\) against \(H_{1}: \theta=0.07\). Show that this test can be based upon the statistic \(\sum_{1}^{n} X_{i}\). If \(\alpha_{a}=0.20\) and \(\beta_{a}=0.10\), find \(c_{0}(n)\) and \(c_{1}(n)\)

Consider a normal distribution of the form \(N(\theta, 4)\). The simple hypothesis \(H_{0}: \theta=0\) is rejected, and the alternative composite hypothesis \(H_{1}: \theta>0\) is accepted if and only if the observed mean \(\bar{x}\) of a random sample of size 25 is greater than or equal to \(\frac{3}{5}\). Find the power function \(\gamma(\theta), 0 \leq \theta\), of this test.

Illustrative Example \(8.2 .1\) of this section dealt with a random sample of size \(n=2\) from a gamma distribution with \(\alpha=1, \beta=\theta\). Thus the mgf of the distribution is \((1-\theta t)^{-1}, t<1 / \theta, \theta \geq 2 .\) Let \(Z=X_{1}+X_{2} .\) Show that \(Z\) has a gamma distribution with \(\alpha=2, \beta=\theta\). Express the power function \(\gamma(\theta)\) of Example 8.2.1 in terms of a single integral. Generalize this for a random sample of size \(n\).
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