\documentclass[12pt]{report} \usepackage{amsmath,amsfonts,amssymb} \setlength{\textwidth}{18cm} \setlength{\textheight}{21cm} \setlength{\topmargin}{-1cm} \setlength{\oddsidemargin}{-1cm} \setlength{\evensidemargin}{-1cm} \newcommand{\bq}{\begin{eqnarray*}} \newcommand{\eq}{\end{eqnarray*}} \begin{document} \pagestyle{empty} \begin{center} \Large {\bf STAT-324}\\ {\bf Final Exam}\\ \large {\bf May 11, 2005\\ 12:25-2:25AM}\\ \normalsize Instructor: Moo K. Chung\\ \vspace{2cm} Answer all questions showing all your work neatly.\\ Calculators are allowed. One page note is permitted \vspace{2cm} \large {\bf Name:}\underline{\hspace{10.5cm}}\\ {\bf Student ID:}\underline{\hspace{9cm}}\\ \normalsize \vspace{1cm} Pledge: On my honor, I have neither given nor received\\ unauthorized aid on this examination.\\ \vspace{1cm} \large {\bf Signature:}\underline{\hspace{9cm}}\\ \vspace{2cm} \begin{tabular}{|r|l|c|}\hline Question & Points}\\ \hline 1 & \\ \hline 2 & \\ \hline 3 & \\ \hline 4 & \\ \hline 5 &\\ \hline Total & \\ \hline \end{tabular} \end{center} \newpage You may use the following {\tt R} output for the answering exam problems. \begin{verbatim} > qnorm(c(0.005,0.01,0.025,0.05,0.1)) [1] -2.575829 -2.326348 -1.959964 -1.644854 -1.281552 > qt(0.025,5:11) [1] -2.57 -2.44 -2.36 -2.30 -2.26 -2.22 -2.20 > qt(0.05,5:11) [1] -2.01 -1.94 -1.89 -1.85 -1.83 -1.81 -1.79 > qt(0.01,5:11) [1] -3.36 -3.14 -2.99 -2.89 -2.82 -2.76 -2.71 > qf(0.025,1,5:11) [1] 0.001084778 0.001067109 0.001054611 0.001045308 0.001038115 0.001032389 [7] 0.001027723 > qf(0.05,1,5:11) [1] 0.004344768 0.004273760 0.004223537 0.004186155 0.004157256 0.004134252 [7] 0.004115506 > qf(0.01,1,5:11) [1] 0.0001735012 0.0001706780 0.0001686808 0.0001671942 0.0001660447 [6] 0.0001651297 0.0001643840 \end{verbatim} \begin{enumerate} \item Two players $A$ and $B$ flip a biased coin alternately and the first player to obtain a head wins. The probability of obtaining a head is $p > 0$ at each toss. Suppose $A$ flips the coin first.\\ (a) Find the probability that $A$ wins the game. Derive your result [10pts].\\ \newpage (b) Find the expected number of tosses $A$ must make till $A$ wins. Derive your result [10pts].\\ \vspace{10cm} \item (a) Suppose that $X$ and $Y$ are Bernoulli random variables with parameter $0< p < 1$ such that $\mbox{cov}(X,Y) = 0$. Prove that $X$ and $Y$ are independent [10pts].\\ \newpage (b) Let $X_1,\cdots,X_n$ be a random sample from the Bernoulli distribution with parameter $p$. Find an unbiased estimator for $(1-p)^2$ [10pts]. \vspace{7cm} \item 47 out of 102 doctors knew the generic name for the drug methadone. Compute a $98\%$ confidence interval for the propotion of all doctors who {\bf do not know} the generic name for methadone. [10 pts]. \newpage \item We wish to fit observations $(x_1,y_1)$, $\cdots$, $(x_n,y_n)$ with a linear model $Y=c x + \epsilon$. We assume that $\epsilon \sim N(0,\sigma^2)$. \begin{itemize} \item[(a)] Estimate $c$ by the least squares estimation. Determine if it is unbiased. Derive your results [10pts]. \vspace{9cm} \item[(b)] Estimate $c$ by the maximum likelihood method and compute the variance of the estimator. Derive your results [15pts]. \newpage \item[(c)] If the sample correlation coefficient of the above paired data is denoted by $r$, what is the sample correlation correlation coefficient of data $(x_1 -\bar x,y_1 -\bar y), \cdots, (x_n -\bar x, y_n -\bar y)$? $\bar x$ and $\bar y$ are the respective sample means of $x_i$'s and $y_i$'s. Prove your result [10pts]. \vspace{11cm} \item[(d)] Prove the sample correlation $r$ is a number between -1 and 1 [10pts]. \end{itemize} \newpage \item Three different types of boxes underwent compression strength test. The following data are measured compression strength (lb). Assume the normality of data. $$\begin{array}{ccccc} \mbox{Type I } &65 & 79 & 72 & 68 \\ \mbox{Type II} & 78 & 77 & 68 & 77 \\ \mbox{Type III} & 60 & 62 & 74 & & \\ \end{array}$$ The following {\tt R} output may be used to answer the following questions. \begin{verbatim} > type1<-c(65,79,72,68) > type2<-c(78,77,68,77) > type3<-c(60,62,74) > z<-c(type1,type2,type3) > w<-c(0,0,0,0,1,1,1,1,2,2,2) > summary(lm(z~w)) Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 73.158 3.092 23.663 2.05e-09 *** w -2.474 2.564 -0.965 0.36 Residual standard error: 6.738 on 9 degrees of freedom Multiple R-Squared: 0.09376, Adjusted R-squared: -0.006933 F-statistic: 0.9312 on 1 and 9 DF, p-value: 0.3598 \end{verbatim} (a) Test if the true compression strength for box types I and III are equal at $\alpha =0.1$ Assume the {\bf equal} variance for different box types. Clearly write down the null and alternate hypotheses, the test statistic, and the statistical model you are using [10pts].\\ \newpage (b) Test if the true compression strength for all box types I, II and III are equal at $\alpha =0.1$. Assume the {\bf equal} variance for different box types. Clearly write down the null and alternate hypotheses, the test statistic, and the statistical model you are using [20pts].\\ \end{enumerate} \end{document}