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\begin{flushright}
\underline{ID (E.g. ED7843):~~~~~~~~~~~~~~~~~~~~~~~~~~~~~}
\underline{Full Name (Print):~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~}
\end{flushright}
\begin{center}
\textbf{MA 2210/6150 Midterm II (11/16, 5:30-7:20p)}
\end{center}

\noindent - Print your full name on \textbf{the first sheet} and at least first or last name
on all the other sheets you will submit.

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

\noindent - Write down the process to the answer EXCEPT FOR Q1 and Q4.
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 (44 points;
a correct answer = 4 pts each, ``Skip'' = 2 pts each, a wrong answer = 0 pts each).

\begin{description}
  \item[(1)] Let $X_1, X_2 \sim Bin(1,p)$, and $X_1$ and $X_2$ are independent.
              Then $X_1 + X_2 \sim Bin(2,2p)$.
  \item[(2)] Assume $0 < p <1$. A binomial distribution is symmetric if and only if $p=0.5$.
  \item[(3)] A Poisson distribution with $\mu=3$ is maximized at $k=2$ and $3$.
  \item[(4)] Let $X \sim N(0,1)$ then $P(X = 0) = 1/2$. 
  \item[(5)] Let $X \sim N(0,1)$ and $a >0$, then $P(-a \le X \le 0) + P(a < X) = 1/2$. 
  \item[(6)] Let $X_1 \sim N(\mu_1, \sigma_1^2)$ and $X_2 \sim N(\mu_2, \sigma_2^2)$ then
               $X_1 - X_2 \sim N(\mu_1 - \mu_2, \sigma_1^2 - \sigma_2^2)$.
  \item[(7)] Let $X_1$ and $X_2$ be independent random variables, 
              and the distribution of $X_1$ and $X_2$ are the same.
               Then the shape of the distribution of $X_1$ and the shape of the distribution
               of $\frac{X_1+X_2}{2}$ are the same, even though the variances are different.
  \item[(8)] Let $X_i$ ($i=1,2,\cdots,n$) be IID random variables with $Var(X_i) = 1$,
               then the variance of $\bar{X} = \frac{X_1 + \cdots + X_n}{n}$ is $1/n$.
  \item[(9)] Let $X \sim Bin(n,p)$, then the variance of $\frac{X}{n}$ goes to zero
               when $n$ gets larger.
  \item[(10)] If a point estimator $\hat{\mu}$ is biased,
               it is never consistent.
  \item[(11)] A confidence interval $[A,B]$ for a parameter $\mu$ is (a pair of) random variables
              (before observing a sample).
\end{description}
\begin{table}[htbp]
 \begin{center}
  \begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|}
    \hline
    (1)   & (2) & (3) & (4) & (5) & (6) & (7) & (8) & (9) & (10) & (11) \\
    \hline
    Yes    & Yes  & Yes  & Yes    & Yes    & Yes    & Yes  & Yes  & Yes  & Yes & Yes \\
    No     & No   & No   & No     & No     & No     & No   & No   & No   & No  & No \\
    Skip   & 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.} (12 points) \\

Suppose that the probability that a college student becomes
a victim of a violent crime during a given year is 0.002.
Let us randomly choose 1,500 college students,
and let $X$ be the number of students who will be victims in a year.
\begin{description}
  \item[(a)] Write down $P(X=k)$ as a function of $k$ (note: we want the exact distribution, not an approximation). (2)
  \item[(b)] Calculate $P(X \le 1)$ to four places of decimals (e.g., 0.1235).  (4)
  \item[(c)] The distribution in (a) can be approximated by a Poisson distribution.
              Write down the probability function $P(X=k)$ of THIS Poisson distribution. (2)
  \item[(d)] Approximate $P(X \le 1)$ by using (c) to four places of decimals. (4)
\end{description}

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

\noindent \textbf{3.} (9 points) \\
\begin{description}
  \item[(a)] Let $Z \sim N(0,1)$. Calculate $P(-0.3 \le Z \le 1.3)$. (3)
  \item[(b)] Let $X \sim N(-3,4^2)$. Calculate $P(-10 \le X \le 5)$. (4)
  \item[(c)] Let $f(z)$ be the density function of $Z \sim N(0,2^2)$. Calculate $f(0)$. (2)
\end{description}

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

\noindent \textbf{4.} (13 points; 1 pt each for A-G, 2 pts for H, 4 pts for I) \\
There are three cards: `1', `2' and `3'.
Let $X$ be the number on the card when we randomly pick up one,
then $EX =$ \fbox{A}. Since 
$E [X^2] = 1^2 \times \frac{1}{3} + 2^2 \times \frac{1}{3} + 3^2 \times \frac{1}{3} = \frac{14}{3}$,
$Var X =$ \fbox{B}.

Next, randomly pick up a card three times with replacement 
(so we are able to pick up the same card more than once).
The average number $\bar{X}$ of the three times is between 1 and 3, and its probability distribution is
in the table below. The variance of $\bar{X}$ is \fbox{C}.
\begin{table}[h]
 \begin{center}
  \begin{tabular}{|c|c|c|c|c|c|c|c|}
    \hline
    average      & 1  & 4/3   & 5/3   & 2  & 7/3  & 8/3 & 3 \\
    \hline
    probability   & $\frac{1}{27}$ & \fbox{D} &  \fbox{E}  &  \fbox{F}  & \fbox{E} & \fbox{D} & \fbox{G} \\
    \hline
  \end{tabular}
 \end{center}
\end{table}

Finally when we randomly pick up a card 600 times with replacement,
the variance of the average number $\bar{X}$ is \fbox{H}.
Since the distribution of $\bar{X}$ is approximately normal by the central limit theorem,
$P(1.9 \le \bar{X} \le 2.1)$ is approximately \fbox{I}.

\begin{table}[h]
{\Large
 \centering
  \begin{tabular}{|l|l|l|}
  %\begin{tabular*}{6.4in}{@{\extracolsep{\fill}}|l|l|l|}
    \hline
    A: ~~~~~~~~~~~~~~~~~~~~~~~~  & B: ~~~~~~~~~~~~~~~~~~~~~~~~  & C: ~~~~~~~~~~~~~~~~~~~~~~~~  \\ [+30pt]
    \hline
    D: ~~~~~~~~~~~~~~~~~~~~~~~~  & E: ~~~~~~~~~~~~~~~~~~~~~~~~  & F: ~~~~~~~~~~~~~~~~~~~~~~~~  \\ [+30pt]
    \hline
    G: ~~~~~~~~~~~~~~~~~~~~~~~~  & H: ~~~~~~~~~~~~~~~~~~~~~~~~  & I: ~~~~~~~~~~~~~~~~~~~~~~~~~~   \\ [+30pt]
    \hline
  \end{tabular}
}
\end{table}

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

\noindent \textbf{5.} (11 points) \\
A random sample of 121 human body temperatures had a mean of 97.7 degrees
and a standard deviation of 0.9 degrees.
\begin{description}
  \item[(a)] Find the value $a$ such that $P(Z > a) = 0.02$ from a normal distribution table. (2)
  \item[(b)] Estimate the 96\% confidence interval for the population mean of body temperatures $\mu$. (4)
  \item[(c)] When we construct 50 confidence intervals from 50 random samples 
         (you may assume each sample has different 121 observations) in the same way as (b),
         how many of them include $\mu$ in expectation? (2)
  \item[(d)]  When we have 3 confidence intervals in the same way as (c),
          what is the probability that all 3 confidence intervals include $\mu$?
          (Answer three places of decimals, e.g. 0.123.) (3)
\end{description}

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

\noindent \textbf{6.} (11 points) \\
We want to see the effectiveness of two pain relievers: A and B,
and obtained the following data.
\begin{table}[h]
 \begin{center}
  \begin{tabular}{|c|cc|}
    \hline
       & A   & B   \\
    \hline
    sample size                   & 300   & 400   \\
    \hline
    the number of effective cases & 250   & 320   \\
    \hline
    \hline
    true proportion of effective cases & $p_1$   & $p_2$ \\
    \hline
  \end{tabular}
 \end{center}
\end{table}
\begin{description}
  \item[(a)] Let $\hat{p}_1$ be the sample proportion for the effective cases for drug A with sample size 300,
             and $\hat{p}_2$ for drug B with sample size 400.
             Estimate the standard error of $\hat{p}_1 - \hat{p}_2$, based on the above data. (3)
  \item[(b)] Explain why $\hat{p}_1 - \hat{p}_2$ is approximately normal. (5)
  \item[(c)] Estimate the 90\% confidence interval for $p_1 - p_2$. (3)
\end{description}

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