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\begin{flushright}
\underline{ID (E.g. ED7843):~~~~~~~~~~~~~~~~~~~~~~~~~~~~~}
\underline{Full Name (Print):~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~}
\end{flushright}
\begin{center}
\textbf{MA 2210/6150 Midterm I (10/12, 5:30-7:20p)}
\end{center}

\noindent - Print your full name on \textbf{all sheets} you will submit.

\noindent - Write down your answers on the same page as the questions. If the space is not enough,
you may use the backside of the sheet.

\noindent - Write down the process to the answer except for multiple choice questions.
You may not get full credits if you only write the final answer.            

\noindent \hrulefill 

\vspace{1zh}

\noindent \textbf{1.} Are the following statements correct? Circle your answers (40 points;
a correct answer = 4 pts each, ``Skip'' = 2 pts each, a wrong answer = 0 pts each).

\begin{description}
  \item[(1)] One survey asked customers to rate a movie.
         Possible choices for customers are: 1(poor),
         2(average), 3(good) and 4(excellent).
         This rating is a discrete variable.
  \item[(2)] The mean is always larger than or equal to the 1st quartile. 
  \item[(3)] The maximum is always larger than or equal to the mean. % \textbf{(Yes/No)}
  \item[(4)] The variance of a sample is always larger than the standard deviation of the sample.
  \item[(5)] When the correlation coefficient is zero, it means there are no
         relationships between the two variables. % \textbf{(Yes/No)}
  \item[(6)] Suppose the correlation coefficient for $(x_1,y_1), \cdots, (x_n,y_n)$ is 0.3.
              Then the correlation coefficient for $(x_1,2 y_1), \cdots, (x_n,2 y_n)$ is 0.6.
  \item[(7)] For any events $A$ and $B$, $P(A \cap B) \le P(B)$.
  \item[(8)] For any random variable $X$ and a constant $a \ge 0$, $\Var(aX) = a \Var(X)$.
  \item[(9)] A binomial distribution is skewed to the right if $1 > p > 0.5$.
  \item[(10)] For any random variable $X$, the expectation of $X^2$ is always larger than or 
          equal to the square of $E[X]$. % \textbf{(Yes/No)}
\end{description}
\begin{table}[htbp]
 \begin{center}
  \begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|}
    \hline
    (1)   & (2) & (3) & (4) & (5) & (6) & (7) & (8) & (9) & (10) \\
    \hline
    Yes    & Yes  & Yes  & Yes    & Yes    & Yes    & Yes  & Yes  & Yes  & Yes \\
    No     & No   & No   & No     & No     & No     & No   & No   & No   & No  \\
    Skip   & Skip & Skip & Skip   & Skip   & Skip   & Skip & Skip & Skip & Skip \\
    \hline
  \end{tabular}
 \end{center}
\end{table}

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\begin{flushright}
\underline{Full Name (Print):~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~}
\end{flushright}

\noindent \textbf{2.} (6 points) \\
We collected 20 pennies and recorded their ages
($:=$ 'Current Year' - `Year on Penny'). The data are:
5, 1, 9, 1, 2, 20 ,0 ,25, 0, 17, 1, 4, 4, 3, 0, 25, 3, 3, 8, 28.
\begin{description}
  \item[(a)] Calculate the median. (2)
  \item[(b)] Draw a relative frequency histogram (Make the first bin include 0-4 (years),
    the second include 5-9 (years) and so on). (4)
\end{description}

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\newpage 
\begin{flushright}
\underline{Full Name (Print):~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~}
\end{flushright}

\noindent \textbf{3.} (18 points) \\
Given the following $(x,y)$ values
\begin{table}[h]
 \begin{center}
  \begin{tabular}{c|cccccc}
    \hline
     x  &  1  &  3  & 6   & 5   & 2   & 7  \\
    \hline
     y   & 5   & 4   & 4   & 2   & 7   & 2   \\
    \hline
  \end{tabular}
 \end{center}
\end{table}
\vspace{-1zh}
\begin{description}
  \item[(a)] Calculate the mean of $x$, sample S.D. of $x$, mean of $y$ and sample S.D. of $y$. (6)
  \item[(b)] Calculate the correlation coefficient between $x$ and $y$. (5)
  \item[(c)] Calculate Regression (least-square) line. (5)
  \item[(d)] If we get one more observation with $x = 4.5$, what is the estimate for $y$? (2)
\end{description}

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\newpage 
\begin{flushright}
\underline{Full Name (Print):~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~}
\end{flushright}


\noindent \textbf{4.} (13 points) \\
In a customer center of the orange computer company,
the number of calls ($X$) received in each minute is described as below.
\begin{table}[h]
 \begin{center}
  \begin{tabular}{c|ccccccc}
    \hline
    \# calls $x$ & 0 & 1 &  2 & 3 & 4 & 5 & 6  \\
    \hline
    probability $p(x)$ & 0.26 & 0.40 & 0.15 & 0.10 & 0.06 & 0.02 & 0.01 \\
    \hline
  \end{tabular}
 \end{center}
\end{table}
\begin{description}
  \item[(a)] Calculate the expectation of $X$ ($EX$) and the standard deviation of $X$ ($\sg_X$). (5)
  \item[(b)] Find the probability that $X$ falls into the interval $\mu \pm 2 \sg$. (4)
  \item[(c)] The customer center can answer 4 calls in a minute at most
              (so it ignores the 5th and 6th calls). Let $Y$ be the number of calls the center answers in a minute.
              Calculate $EY$ (expectation of $Y$). Is $EY$ larger than $EX$? (4)
\end{description}

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\newpage 
\begin{flushright}
\underline{Full Name (Print):~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~}
\end{flushright}

\noindent \textbf{5.} (16 points) \\
There is a deck of playing cards (i.e., there are 52 cards;
13 of each suit spades, hearts, diamonds and clubs. Each suit has
A, 2, 3, $\cdots$, 10, J, Q and K).
\begin{description}
  \item[(a)] When you randomly choose two cards, what is the probability
              that both cards are spades? (3)
  \item[(b)] You are going to randomly pick up one card out of 52.
              Let $A$ be the event that the card is a face card (J, Q or K),
              and $B$ be the event that the card is black (spades or clubs).              
              Prove $A$ and $B$ are independent. (6)
  \item[(c)] Now we only use the 13 spade cards ($A,2,3,\cdots,10,J,Q,K$). 
              A child draw one of them, and said ``this is a face card'' (she can not read letters).
              Then you pick up a card from the remaining 12.
              What is the (conditional) probability that your card is a King. (7)      
\end{description}

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\newpage 
\begin{flushright}
\underline{Full Name (Print):~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~}
\end{flushright}

\noindent \textbf{6.} (7 points) \\
Suppose that 60\% of all adults in US prefer Pepsi to Coke
when asked to state a preference.
A group of 150 adults were randomly selected and their preferences recorded.
\begin{description}
  \item[(a)] Calculate the expectation and standard deviation of the number of people who like Pepsi. (4)
  \item[(b)] What is the probability that between 84 and 96 people prefer Pepsi
             (use an empirical rule for normal histogram). (3)
\end{description}

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