Chapter 8

Let $X_1,X_2,\dots ,X_{25}$ denote a random sample of size 25 from a normal distribution $N(\theta ,100)$. Find a uniformly most powerful critical region of size $\alpha =0.10$ for testing $H_0:\theta =75$ against $H_1:\theta >75$

Let $X_1,X_2,\dots ,X_n$ denote a random sample from a normal distribution $N(\theta ,100).$ Show that $\text{Missing left or extra right}$ is a best critical region for testing $H_0:\theta =75$ against $H_1:\theta =78$. Find $n$ and $c$ so that $$P_{H_0}[(X_1,X_2,\dots ,X_n)\in C]=P_{H_0}(\overline{X}\ge c)=0.05$$ and $$P_{H_1}[(X_1,X_2,\dots ,X_n)\in C]=P_{H_1}(\overline{X}\ge c)=0.90$$

Let $X_1,X_2,\dots ,X_n$ denote a random sample from a normal distribution $N(\theta ,16)$. Find the sample size $n$ and a uniformly most powerful test of $H_0:\theta =25$ against $H_1:\theta <25$ with power function $\gamma (\theta )$ so that approximately $\gamma (25)=0.10$ and $\gamma (23)=0.90$.

If $X_1,X_2,\dots ,X_n$ 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 $H_1:\theta =2$

Let \(Y_{1}

Let $X_1,X_2,\dots ,X_n$ be iid with pmf $f(x;p)=p^x(1-p{)}^{1-x},x=0,1$, zero elsewhere. Show that $\text{Missing left or extra right}$ 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 $P_{H_0}(\sum _1^n{X}_i\le c)=0.10$ and $P_{H_1}(\sum _1^n{X}_i\le c)=0.80$.

Consider a distribution having a pmf of the form $f(x;\theta )={\theta }^x(1-\theta {)}^{1-x},x=$ 0,1, zero elsewhere. Let $H_0:\theta =\frac{1}{20}$ and $H_1:\theta >\frac{1}{20}.$ Use the central limit theorcu? to determine the sample size $n$ of a random sample so that a uniformly most powerful test of $H_0$ against $H_1$ has a power function $\gamma (\theta )$, with approximately $\gamma \left(\frac{1}{20}\right)=0.05$ and $\gamma \left(\frac{1}{10}\right)=0.90$.

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 $(1-\theta t{)}^{-1},t<1/\theta ,\theta \ge 2.$ 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 $n.$

A random sample $X_1,X_2,\dots ,X_n$ arises from a distribution given by $$H_{0}: f(x ; \theta)=\frac{1}{\theta}, \quad 0

Let $X_1,X_2,\dots ,X_{10}$ denote a random sample of size 10 from a Poisson distribution with mean $\theta .$ Show that the critical region $C$ defined by $\sum _1^{10}{x}_i\ge 3$ 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 $\theta =0.5$.