\documentclass{uwaexam} \usepackage{uwamaths} \begin{document} %%\title{First Semester Examinations} %%\date{June 1998} %%\deptandcode{Mathematics}{539} \coursenum{170} %\coursenum{175} %%%6 pages %%%10 questions %\duration{Three hours} \maketitle \begin{instructions} The exam consists of 5 Statistics questions and 5 Calculus questions. All questions may be attempted. Each question is worth 10 marks. \end{instructions} \begin{questions} \item% \begin{subq} \item The sample mean and variance for a set of data $x_1,x_2,\dots,x_n$ are given by \[\bar x = \frac{1}{n} \sum^n_{i = 1} x_i \qquad\text{and}\qquad s^2_x = \frac{1}{(n - 1)} \sum^n_{i=1} (x_i - \bar x)^2 \] respectively. If a linear transformation $u_i = ax_i + b$, $i = 1$, $2$, \dots, $n$, is made of the data, show that \begin{subsubq} \item $\bar u = a\bar x + b$ and %1mks \item $s^2_u = a^2 s^2_x$. %2mks \item If $s^2_x = 9$ and $a = -4$, calculate the standard deviation of the $u _i$'s. %1mks \end{subsubq} \item A sample of size $n$ is taken from a population of size $N \geq n$. Write a sentence or two explaining what is meant by the following \begin{subsubq} \item The sample is obtained by \textbf{simple random sampling}. %1mks \item $N = kn$ for some integer constant $k > 1$ and the sample is obtained by \textbf{systematic sampling}. %1mks \end{subsubq} %\newpage \item Let $X_1,\dots,X_n$ be independent and identically distributed random variables representing $n$ measurements taken on a population characteristic whose mean is $\mu_X$. Show that the sample mean $\bar X = \frac{1}{n} \sum^n_{i = 1}\limits X_i$ is an \textbf{unbiased estimator} of $\mu_X$, and explain what is meant by the term in \textbf{bold} type. %4mks \end{subq} \item% \begin{subq} \item Define the following concepts: \begin{subsubq} \item The \textbf{sample space} $S$ of an experiment; \item An \textbf{event}; \item A \textbf{probability function} (or \textbf{measure}) on $S$ (be sure to specify the conditions it must satisfy); \item A \textbf{random variable} on $S$. %4mks \end{subsubq} \item A rare disease $D$ has an incidence of 0.1\%. If a person has $D$ then a diagnostic test signals that fact with probability $0.99$ (the sensitivity). However on a disease free person, the test indicates absence of the disease with probability $0.98$ (the specificity). \begin{subsubq} \item A person is selected at random and tested. What is the probability that the test gives a positive result? %2mks \item A person tests positive. Find the probability that he/she actually has $D$. %4mks \end{subsubq} \end{subq} \item A spinner for a board game has five sides, numbered 1 to 5. So a single spin gives one of these numbers with equal probability. Suppose it is given two independent spins, let $X$ be the score from the first spin, and let $Y$ be the number of odd-numbered scores from the two spins. Thus $X$ and $Y$ are random variables. \begin{subq} \item Determine the joint probability mass function of $X$ and $Y$, displaying it in the usual tabular form. %3mks \item Calculate $E(X)$, $var (X)$, $E(Y)$, and $cov (X, Y)$. %5mks \item Are $X$ and $Y$ independent random variables? Justify your answer. %2mks \end{subq} \item% \begin{subq} \item A random variable $X$ has the probability density function \[f(x) = \begin{cases} cx &\text{if }0 \leq x \leq 2 \\ 0 &\text{otherwise,} \end{cases} \] where $c$ is a positive constant. \begin{subsubq} \item Calculate $c$. %1mks \item Calculate $E(X) $ and $E(1 / X)$. %2mks \item Define and calculate the cumulative distribution function of $X$. %2mks \end{subsubq} \item From past experience the Taxation Office finds that in 10\% of tax returns, the claims for a particular kind of deduction are invalid. The Office believes that invalid claims are more frequent for a particular occupation group. In a random sample of 20 tax returns from this group, five invalid claims are found. \begin{subsubq} \item Write down suitable null and alternative hypotheses to test this belief. %2mks \item Compute the $p$-value for a test of the null hypothesis, and hence decide at the 5\% level whether or not the belief is justified. %3mks \end{subsubq} \end{subq} \item% \begin{subq} \item Transaction times (defined as the times spent queueing and processing a user's requests) at an automatic teller machine (ATM) in a particular bank can be modelled as random variables which have a normal distribution with mean 270 seconds and standard deviation $23.4$ seconds. \begin{subsubq} \item An ATM customer is picked at random. What is the probability that the customer's transaction time will exceed 300 seconds. %1mks \item 20 customers are chosen at random and $N$ is the number whose transaction time exceeds 300 seconds. State the distribution of $N$. %1mks \item Determine the probability that no more than 3 out of 20 customers will experience transaction time over 300 seconds. %1mks \end{subsubq} \item The ATM dealer claims a software upgrade will reduce transaction times. The software is installed, and the observed transaction times of 50 customers are entered into the column $\texttt{C1}$ of MINITAB. The bank's statistical consultant obtained the following print out. \bigskip \begin{flushleft} \textbf{T-Test of the Mean} \begin{verbatim} Test of mu = 270.00 vs mu < 270.00 Variable N Mean StDev SE Mean T P-Value C1 50 265.48 17.47 2.47 -1.83 0.037 \end{verbatim} \end{flushleft} \bigskip \item State the mean and variance of the 50 transaction times.%1mks \item Formulate the hypothesis the statistician should test, and the alternative hypothesis. %1mks \item Give a formula for a statistic designed to test the hypothesis, and state the distribution of this statistic under the assumption that the observations have a normal distribution. %2mks \item Give the observed value of the statistic in (iii) for the transaction data. %1mks \item State with reasons whether or not the hypothesis is rejected at the 5\% significance level, and interpret your result. %1mks \item Compute a 95\% confidence interval for the population mean of transaction times, $\mu$. %1mks \end{subq} \item Let $f(x)=\dfrac{x}{1+x^2}$. \begin{subq} \item Draw a sign diagram for $f'(x)$. %2mks \item Show $f''(x)=\dfrac{2x(x^2-3)}{(1+x^2)^3}$ and hence draw a sign diagram for $f''(x)$. %3mks \item Evaluate $\displaystyle{\lim_{x\to +\infty}}f(x)$ and $\displaystyle{\lim_{x\to -\infty}}f(x)$. %2mks \item Sketch the graph of $y=f(x)$, and label with $(x,y)$-coordinates each point of relative maximum and minimum, and each point of inflection. %3mks \end{subq} \item Let $f(x)=2x-1+\sin x.$ \begin{subq} \item Show that $f'(x)>0$ for all $x$, and that the equation $f(x)=0$ has exactly one solution. %2mks \item By calculating $f(0)$ and $f(0.5)$, show the solution of $f(x)=0$ lies between $0$ and $0.5$. %2mks \item Solve $f(x)=0$ to 4 decimal place accuracy, using 3 iterations of the Newton-Raphson algorithm, starting at $x_0=0.5$. %6mks \end{subq} \item \begin{subq} \item During a certain 12 hour period, the temperature at time $t$ hours after the start of the period, was $15+4t-\frac{1}{3}t^2$ degrees Celsius. \begin{subsubq} \item Show that the temperature was the same at the beginning as at the end of the period. Briefly, how did the temperature change during the period? %1mks \item What was the average temperature during that period? %3mks \end{subsubq} \item The annual world rate of water use $t$ years after 1940, for $0\le t\le 40$, was approximately $860e^{0.04t}\,\text{km}^3/\text{yr}$. \begin{subsubq} \item How much water was used between 1940 and 1980? %2mks \item Express your answer to part (a) in kilolitres, given that $1\,\text{litre} = 1000\,\text{cm}^3$.%2mks \item By what year had the rate of water use doubled since 1940? %2mks \end{subsubq} \end{subq} \item Let $f(x,y) = y^3-x^2+6x-12y+5$. \begin{subq} \item Find the relative minimum and relative maximum points of $f(x,y)$, and determine whether each is a minimum or a maximum. %8mks \item What conclusions can you draw about the absolute maximum and minimum of $f(x,y)$? %2mks \end{subq} \item% \begin{subq} \item Solve the differential equations: \begin{subsubq} \item $y'=\left(\dfrac{e^x}{y}\right)^2$. %2mks \item $y\,y'=t\cos(t+1)$. %3mks \end{subsubq} \item The Gompertz growth equation for biological populations is \[\frac{dy}{dt} = -ay\ln\frac{y}{b}\] where $a$ and $b$ are positive constants, i and $y(t)$ is the population at time $t$. Find the general form of the solutions to this equation, and discuss their qualitative behaviour. %5mks \end{subq} \end{questions} \end{document}