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\begin{center}
\Huge \textbf{Exercise 8}
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\begin{enumerate}
\item Suppose that $X_1,...,X_n$ and $Y_1,...,Y_n$ are two independent
random samples from two exponential distributions with mean $\mu_1$ and $\mu_2$
respectively.
Find the likelihood ratio test for
$H_0: \mu_1 = \mu_2$ against $H_1: \mu_1 \ne \mu_2$.
Specify the asymptotic distribution of the test statistic under $H_0$.


\item
A survey of the use a particular product was conducted in four areas, and a
random sample of 200 potential users was interviewed in each area. In area $i$,
for $i=1,2,3,4,$ $x_i$ of the 200 said that they used the product. Construct
a likelihood ratio test to test whether the proportion of the population
using the product is the same in each of the four areas. Carry out the test
at 5\% level for the case $x_1=76, x_2=53, x_3 =59$ and $x_4=48$.

\item
A random sample $X_1, \cdots, X_n$ of size $n$ is selected from a normal distribution
with known mean $\mu$ and unknown variance $\sigma^2$. Two possible confidence
intervals for $\sigma^2$ are shown below, where
$a_1, a_2, b_1$ and $b_2$ are constants.
\[
 \;(a_1^{-1}\sum_{i=1}^n (X_i - \bar X)^2, \; a_2^{-1}\sum_{i=1}^n (X_i - \bar X)^2),
\quad \quad
 \;(b_1^{-1}\sum_{i=1}^n (X_i - \mu )^2, \; b_2^{-1}\sum_{i=1}^n (X_i - \mu )^2).
\]
For the case $n=10$, find values of these constants which give intervals with
confidence level 0.90. Compare the expected lengths of these intervals.
Comment on your findings.


\item
Let $X_1, \cdots, X_n$ be a random sample
from the uniform distribution on the interval
$[0, \theta]$ $(\theta >0)$. Find a confidence interval for $\theta$.


\end{enumerate}
