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\begin{document}





\author{Brian C. Hall}

\title{Math 214--Test 2}

\date{Wednesday, April 11, 2001}

\maketitle







\textbf{Instructions}. There are 7 problems worth 14 points each, with 2 free

points, for a total of 100 points possible. Answer all the questions.

\textit{Show your work on all problems.} Calculators are permitted. Notes and

books are not permitted. You have one hour for the test.



This test is subject to the Notre Dame honor code.\bigskip



\textbf{Name}:\bigskip



\begin{enumerate}

\item  A magazine publisher claims that its subscribers have an average salary

of \$65,000. To test this claim, a survey is conducted of 200 randomly chosen

subscribers. In the survey, the average salary is \$62,420 with a standard

deviation of \$12,290. Perform a hypothesis test of the publisher's claim.

State the null hypothesis and the alternative hypothesis. Do the results of

the survey give sufficient evidence to dispute the publisher's claim, at the

1\% significance level?\pagebreak 



\item  Someone plays the lottery 300 times in the course of one year. Each

ticket costs \$1, a winning ticket is worth \$10, and the probability of

winning on any one ticket is 1/12. What is the probability that the person

will make a profit?\pagebreak 



\item  A survey of 395 randomly chosen registered voters in a given county

shows 182 who consider themselves liberals. Construct a 95\% confidence

interval for the proportion of registered voters in the county who consider

themselves liberals.\pagebreak 



\item  A machine is supposed to produce at most 5\% defective parts. For

quality control, a random sample of $n$ parts is chosen, and if the proportion

of defective parts in the sample exceeds 8\%, the machine is recalibrated.



a) How large should the sample size $n$ be so that if the machine is actually

producing 5\% defective parts, there is only a 2\% chance of a recalibration?



b) Using the sample size determined in part (a), what is the probability of a

recalibration if the machine is actually producing 10\% defective parts?\pagebreak 



\item  A study surveys 250 randomly chosen women ages 30-35 and 240 randomly

chosen men ages 30-35. The heights of the participants are measured with the

following results.

\[%

\begin{array}

[c]{ccc}%

\text{women} &  & \text{men}\\

\bar{X}=64.2\text{ inches} &  & \bar{X}=68.9\text{ inches}\\

s=3.2\text{ inches} &  & s=4.0\text{ inches}%

\end{array}

\]

Construct a 97\% confidence interval for the difference between the average

height of men and the average height of women ages 30-35 in the population.\pagebreak 



\item  Consider the function $f(x)$ given by

\[

f\left(  x\right)  =\left\{

\begin{array}

[c]{cc}%

2\frac{1}{x^{3}} & x\geq1\\

0 & x<1

\end{array}

.\right.

\]



a) Show that $f(x)$ is a valid probability density function.



b) If $X$ is a continuous random variable with probability density function

$f(x),$ compute the mean of $X.$\pagebreak 



\item  A randomized, placebo-controlled study is conducted to see whether a

new drug reduces the recurrence of strokes in women over the age of 65. The

study considers 1025 women over 65 who have had a stroke. They are randomly

divided into two groups, with 512 to receive the drug and 513 to receive a

placebo. The women are followed for two years and the number suffering a

second stroke is recorded. The results are that 59 women taking the drug and

80 women taking the placebo have a second stroke. Does the study indicate that

the drug is effective? Use the 1\% significance level.

\end{enumerate}

\end{document}