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\hskip 1.5in \textbf{Midterm 3B--Stat 100--Spring 1997}   
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You may use calculators, and ONE page of notes (writing allowed on 
both sides). 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. 
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1.
In a sample of 100 bottles of a new brand of beer, 
the mean alcohol content was  5.1\% with 
standard deviation  0.5\% .  Does this data
provide strong evidence that the average alcohol content for
this brand is different from 5\%? To this end,
\begin{enumerate}
\item
Formulate the null and the alternative hypotheses. 
\item 
Define the appropriate test statistic.  
\item 
Give the form
of the decision procedure at level of significance $\alpha$.
\item
Calculate the P-value for the sample mean found above and determine
for which $\alpha$ the null hypothesis is retained. 
\end{enumerate}

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2.
Suppose we have  the following random sample of 6 observations from a normal
population: 
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16, 14, 12, 8, 10, 6
\end{center}

For this data, the sample mean is 11 and the 
sample standard deviation is $\sqrt{14} \approx 3.742$ .
\begin{enumerate}
\item
Give a point estimate and a 98\%-confidence interval
for the population standard deviation
$\sigma$.
\item
Do the above observations provide strong evidence that $\sigma \neq 3.5$?
Test at $\alpha=0.02$.
\end{enumerate}

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3. In a random sample of 100 New York residents, the average weekly
salary is $\$510$ and the standard deviation is $\$100$.  
In a random sample of 200 Chicago residents, the average weekly salary 
is $\$450$ and the standard deviation is $\$85$.  
Let $\mu_N$ be the average weekly
salary of New York residents and $\mu_C$ be the average weekly salary
of Chicago residents.

Find a $90\%$ confidence interval for $\mu_C-\mu_N$.

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**************THERE ARE MORE PROBLEMS ON THE BACK SIDE***************
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\hspace{6in} .  
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\hspace{2.5in} IN THE PROBLEMS BELOW: 

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If your answer involves a complicated formula, you need not take 
the time to evaluate it on your calculator; just write it out with 
the appropriate numbers inserted. 

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\hspace{2.5in} ------------------------------------

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4. A farmer has paired 5 adjacent plots of land and is going to plant
green beans.  On the first plot, the farmer has used the fertilizer
``Grow-ever.''  On the second plot, 
the farmer has used the fertilizer ``Healthy-Soil.''  The
number of gallons of green beans 
produced on each plot of land is given below. 
\vspace{.1in} 

\hspace{.2in} 
\hspace{1.2in}  1 \hspace{.3in}2 \hspace{.3in}3 \hspace{.3in}4 \hspace{.3in}5

\hspace{.2in} 
Grow-ever \hspace{.4in} 50 \hspace{.2in}58 \hspace{.2in}60
\hspace{.2in}65 \hspace{.2in}42

\hspace{.2in} 
Healthy-Soil \hspace{.26in} 45 \hspace{.2in}60 \hspace{.2in}50
\hspace{.2in}63 \hspace{.2in}37

\vspace{.1in}

Test the claim that using the fertilizer ``Grow-ever''  results in a
higher yield of green beans than using ``Healthy-Soil.''  Use a $5\%$
significance level.  In your response,

\begin{enumerate}
\item
Define the null and alternative hypotheses.
\item
Define the test statistic and rejection region. 
\item 
Determine the value of the test statistic.
\item
Decide if the data support the claim that ``Grow-ever'' is better
than ``Healthy-Soil'' for growing green beans.  
\end{enumerate}
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5. At a certain factory, a time study of two assembly methods is conducted,
  in which nine  workers 
use method 1 and eleven workers use method 2. 
The means and standard deviations of the assembly times are 
 as follows:

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 \hspace{2in} Method 1 \hspace{.5in} Method 2

\hspace{.2in} mean	\hspace{1.43in}    77.0  \hspace{1in} 85.1

\hspace{.2in} standard deviation \hspace{.55in}   4.1  \hspace{.98in}  9.2

\vspace{.1in}
\begin{enumerate}
\item (14 pts.)
Compute a conservative $95$\% confidence interval for $\mu _1 -\mu_2$, 
the mean 
difference  in assembly time for the two methods at the factory.  
\item (3 pts.) 
In what way is a {\it conservative} $95$\% confidence interval 
for $\mu _1 -\mu_2$ different from a $95$\% confidence interval 
for $\mu _1 -\mu_2$ ?
\item (2 pts.) 
Briefly indicate any assumption 
 you make to justify this procedure. 
\item (3 pts.) 
Would you be justified in pooling to compute a $95$\% confidence interval? 
Briefly explain. 
\end{enumerate}
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