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\hskip 1.5in \textbf{Midterm 1A--Stat 100--Spring 1997}   
\vskip .2in  
You may use calculators, but not books or notes. Each problem is 
worth 20 points. Different parts of a problem have equal weight unless 
otherwise indicated.  Do not spend too much time on any one problem. 
 Put a box around the final answer to a question. 
\vskip .1in 
1.  (a) (15 pts.) For the data  $\  2, 5, 5, 2, 3 \ $, 
calculate 
the sample mean,  the 
sample standard deviation, and  the 
median .  
\newline 
(b) (5 pts.) Suppose that data $x_1, x_2, \dots x_n$ has sample 
mean equal to 6.  What is the sample mean of the data 
$\ x_1 + 3,\  x_2+3,\  \dots\  x_n +3\ $ ? 
\newline 
\newline 
 2.  Let the sample space $S = \{e_1,e_2,e_3,e_4,e_5,e_6\}$.  
Suppose $P(e_1)=P(e_2)=.2$ and $P(e_3)=P(e_4)=.05$, and suppose $e_5$
and $e_6$ are equally likely.  Consider the events
$A=\{e_1,e_4,e_5\},\
B=\{e_2,e_4,e_5,e_6\}.$


a.  Find $P(e_5)$.

b.  Find $P(A \overline{B})$.

c.  Find $P(A|B)$.

d.  Are $A$ and $B$ independent?  Why or why not?
\newline \newline
3.  From a group of 25 cities, 4 will be randomly selected
 for a new pollution study.  Of the 25 cities,  4 have
 traditionally had high pollution, 15 have
had moderate pollution, and 6 have had low 
pollution.

(a) What is the probability that 4 high-pollution cities will 
be selected? 

(b)  Find the probability that 2 low-pollution cities 
and 2 high-pollution cities will be selected.
\newline 
\newline 
4. For cars visiting a certain county park, assume  that 
X, the number of people in a car,  
has the following probability distribution. 
\vspace{10pt}

\begin{center}
\begin{tabular}{|c||c|c|c|c|c|c|} \hline 
$n$ & 1 & 2 & 3 & 4 & 5 & 6 or more \\ \hline
$P([X=n])$ & 0.1 & 0.2 & 0.4 & 0.2 & 0.1 & 0 \\ \hline
\end{tabular}
\end{center}

\renewcommand{\labelenumi}{\alph{enumi}.}
\vspace{10pt}


\begin{enumerate}
\item What is the probability that a car contains 
just one or two people? 

\item 
Compute E(X), the expected number of people in a car.  

\item 
What is  the standard deviation of the
number of people in a car? 

\item 
If park officials plan to admit a certain number of cars with 
the aim of admitting approximately 1200 people, 
then how many cars should they admit to the park? 

\end{enumerate}

\vspace{10pt}

5. Let X denote the face shown by a loaded die,
and assume the probability distribution is

\vspace{10pt}

\begin{center}
\begin{tabular}{|c||c|c|c|c|c|c|} \hline 
$n$ & 1 & 2 & 3 & 4 & 5 & 6  \\ \hline
$P([X=n])$ & 0.2 & 0.2 & 0.3 & 0.1 & 0.1 & 0.1 \\ \hline
\end{tabular}
\end{center}

\vspace{10pt}

Suppose we roll this die 20 times.
Find the probabilities of the following events. 
\vspace{10pt}

\begin{enumerate}
\item
Face \lq\lq 3" appears on both the 10th and 11th rolls.
\item
Face \lq\lq 3" shows up no more than 5 times. 
\item
Face \lq\lq 3" shows up exactly 7 times.
\item 
Face \lq\lq 3" shows up at least 8 times.
\end{enumerate}
 

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From mdm@math.umd.edu Tue Apr  1 14:39:51 1997
Received: from rosza.umd.edu (rosza.umd.edu [129.2.56.36]) by math.umd.edu (8.6.11/8.6.4) with ESMTP id OAA11398 for <mmb@math.umd.edu>; Tue, 1 Apr 1997 14:39:49 -0500
From: Mark D Morgan (student) <mdm@math.umd.edu>
Received: (mdm@localhost) by rosza.umd.edu (8.6.11/8.6.11) id OAA17545 for mmb; Tue, 1 Apr 1997 14:39:40 -0500
Date: Tue, 1 Apr 1997 14:39:40 -0500
Message-Id: <199704011939.OAA17545@rosza.umd.edu>
To: mmb@math.umd.edu
Subject: Test 2 problems
Status: RO


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Version 1

1.  A reporter would like to determine the average commuting time to work
of all people in the metropolitan D.C. area.  The standard deviation
of commuting times is known to be 14 minutes.  The reporter will
collect a sample of commuting times.  How large should the sample be
in order to be $95\%$ sure that the sample mean is within 2 minutes of
actual mean? 

2.  The campus food service is conducting a study concerning the
number of students who eat breakfast on a regular basis.  In a survey
of 250 students, 133 of the students eat breakfast regularly.
Construct a $99\%$ confidence interval for the proportion of students
who eat breakfast regularly.


Version 2

1.  The U.S. Transportation Department would like to estimate the
proportion of traffic accidents which involve people over the age of
70.  This proportion is unknown, and the department will randomly
select traffic accident reports to estimate it.  How many reports
should the department select in order to be $97\%$ confident that the
estimate is within 0.01 of the true proportion?
	      
2.  A group of 300 women over the age of 30 are given a cholesterol
test.  The sum of all of these observations is 52,800.  The sample
standard deviation of the oberservations is 15.  Construct a $90\%$
confidence interval for the actual cholesterol level of women over the
age of 30.     

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