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Name \underline{\hspace{60mm}} \\ 
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Student Number \underline{\hspace{60mm}}
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{\Large \textbf{STA 312 f2023 Quiz 3}}\\
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\noindent
Let $X_1, \ldots, X_n$ be a random sample (that is, independent and identically distributed) from a Poisson distribution with parameter $\lambda>0$. You already know that the maximum likelihood estimate is $\widehat{\lambda} = \overline{X}$. We want to test $H_0: \lambda = \lambda_0$ versus $H_1: \lambda \neq \lambda_0$ with a large-sample likelihood ratio test. For this problem, the subset of the parameter space specified by the null hypothesis is a single point: $\Theta_0 = \{ \lambda_0 \}$.

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\item  (7 points) \label{Gsq} Write down and simplify the $G^2$ test statistic. A variety of ``simplified" answers can be correct. Your final answer is a formula. \textbf{Circle it}.

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\item (3 points) 
        \begin{enumerate}
            \item A random sample of size $n=49$ yields a sample mean of 4.2 and a sample standard deviation of 2.14. We want to test $H_0: \lambda=3$. Calculate your $G^2$ statistic from Question~\ref{Gsq}. Show a little work. The answer is a number. \textbf{Circle your answer}. \vspace{50mm}
            \item What are the degrees of freedom? The answer is a number.  \vspace{5mm}
            \item The critical chi-squared value at $\alpha=0.5$ is $1.96^2 = 3.84$. Do you reject $H_0$? Answer Yes or No. 
        \end{enumerate}



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