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Introduction to Mathematical Statistics

Robert V. Hogg, Allen Craig, Joseph W. McKean

Chapter 8

Optimal Tests of Hypotheses - all with Video Answers

Educators

Section 1

Most Powerful Tests

01:26

Problem 1

In Example $8.1 .2$ of this section, let the simple hypotheses read $H_{0}: \theta=$ $\theta^{\prime}=0$ and $H_{1}: \theta=\theta^{\prime \prime}=-1 .$ Show that the best test of $H_{0}$ against $H_{1}$ may be carried out by use of the statistic $\bar{X}$, and that if $n=25$ and $\alpha=0.05$, the power of the test is $0.999+$ when $H_{1}$ is true.

Manik Pulyani
Manik Pulyani
Numerade Educator
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Problem 2

Let the random variable $X$ have the pdf $f(x ; \theta)=(1 / \theta) e^{-x / \theta}, 0<x<\infty$, zero elsewhere. Consider the simple hypothesis $H_{0}: \theta=\theta^{\prime}=2$ and the alternative hypothesis $H_{1}: \theta=\theta^{\prime \prime}=4$. Let $X_{1}, X_{2}$ denote a random sample of size 2 from this distribution. Show that the best test of $H_{0}$ against $H_{1}$ may be carried out by use of the statistic $X_{1}+X_{2}$

Victor Salazar
Victor Salazar
Numerade Educator
01:34

Problem 3

Repeat Exercise $8.1 .2$ when $H_{1}: \theta=\theta^{\prime \prime}=6$ Generalize this for every $\theta^{\prime \prime}>2$

Manik Pulyani
Manik Pulyani
Numerade Educator
01:22

Problem 4

Let $X_{1}, X_{2}, \ldots, X_{10}$ be a random sample of size 10 from a normal distribution $N\left(0, \sigma^{2}\right) .$ Find a best critical region of size $\alpha=0.05$ for testing $H_{0}: \sigma^{2}=1$ against $H_{1}: \sigma^{2}=2 .$ Is this a best critical region of size $0.05$ for testing $H_{0}: \sigma^{2}=1$ against $H_{1}: \sigma^{2}=4 ?$ Against $H_{1}: \sigma^{2}=\sigma_{1}^{2}>1 ?$

Manik Pulyani
Manik Pulyani
Numerade Educator
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Problem 5

If $X_{1}, X_{2}, \ldots, X_{n}$ is a random sample from a distribution having pdf of the form $f(x ; \theta)=\theta x^{\theta-1}, 0<x<1$, zero elsewhere, show that a best critical region for testing $H_{0}: \theta=1$ against $H_{1}: \theta=2$ is $C=\left\{\left(x_{1}, x_{2}, \ldots, x_{n}\right): c \leq \prod_{i=1}^{n} x_{i}\right\}$.

Victor Salazar
Victor Salazar
Numerade Educator
01:43

Problem 6

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 $H_{1}: \theta_{1}=\theta_{1}^{\prime \prime}=1, \theta_{2}=\theta_{2}^{\prime \prime}=4$.

Manik Pulyani
Manik Pulyani
Numerade Educator
01:43

Problem 7

Let $X_{1}, X_{2}, \ldots, X_{n}$ denote a random sample from a normal distribution $N(\theta, 100) .$ Show that $C=\left\{\left(x_{1}, x_{2}, \ldots, x_{n}\right): c \leq \bar{x}=\sum_{1}^{n} x_{i} / n\right\}$ is a best critical
region for testing $H_{0}: \theta=75$ against $H_{1}: \theta=78$. Find $n$ and $c$ so that
$$P_{H_{0}}\left[\left(X_{1}, X_{2}, \ldots, X_{n}\right) \in C\right]=P_{H_{0}}(\bar{X} \geq c)=0.05$$
and
$$P_{H_{1}}\left[\left(X_{1}, X_{2}, \ldots, X_{n}\right) \in C\right]=P_{H_{1}}(\bar{X} \geq c)=0.90$$

Manik Pulyani
Manik Pulyani
Numerade Educator
01:18

Problem 8

If $X_{1}, X_{2}, \ldots, X_{n}$ is a random sample from a beta distribution with parameters $\alpha=\beta=\theta>0$, find a best critical region for testing $H_{0}: \theta=1$ against $H_{1}: \theta=2$

Manik Pulyani
Manik Pulyani
Numerade Educator
00:55

Problem 9

Let $X_{1}, X_{2}, \ldots, X_{n}$ be iid with pmf $f(x ; p)=p^{x}(1-p)^{1-x}, x=0,1$, zero elsewhere. Show that $C=\left\{\left(x_{1}, \ldots, x_{n}\right): \sum_{1}^{n} x_{i} \leq c\right\}$ is a best critical region for testing $H_{0}: p=\frac{1}{2}$ against $H_{1}: p=\frac{1}{3} .$ Use the Central Limit Theorem to find $n$ and $c$ so that approximately $P_{H_{0}}\left(\sum_{1}^{n} X_{i} \leq c\right)=0.10$ and $P_{H_{1}}\left(\sum_{1}^{n} X_{i} \leq c\right)=0.80$.

Manik Pulyani
Manik Pulyani
Numerade Educator
01:39

Problem 10

Let $X_{1}, X_{2}, \ldots, X_{10}$ denote a random sample of size 10 from a Poisson distribution with mean $\theta .$ Show that the critical region $C$ defined by $\sum_{1}^{10} x_{i} \geq 3$ is a best critical region for testing $H_{0}: \theta=0.1$ against $H_{1}: \theta=0.5 .$ Determine, for this test, the significance level $\alpha$ and the power at $\theta=0.5$.

Manik Pulyani
Manik Pulyani
Numerade Educator