The data for this question relate to air pollution in 41 American  cities.
The variables measured were:

\vspace{2mm}

\begin{center}
\begin{tabular}{ll}
\sf{SO2}: & annual mean $\mbox{SO}_2$ content ($\mu$g per $\mbox{m}^3$). \\
\sf{temp}: & average annual temperature ($^\circ$F). \\
\sf{manu}: & number of manufacturing plants employing $\ge 20$ workers. \\
\sf{pop}: & population size (in thousands). \\
\sf{wind}: & average annual wind speed  (mph). \\
\sf{prep}: & average annual precipitation (inches). \\
\sf{days}: & average number of days with precipitation per year. \\
\end{tabular}
\end{center}

\vspace{5mm}

A regression model that predicts {\sf SO2} was 
obtained using R:
\begin{verbatim}
     > usair.fit<-lm(SO2~.,data=usair.df)
     > summary(usair.fit)
     Coefficients:
                  Estimate Std. Error t value Pr(>|t|)   
     (Intercept) 108.29560   46.18128   2.345  0.02519 * 
     temp         -1.22725    0.60602  -2.025  0.05101 . 
     manu          0.06497    0.01823   3.565  0.00114 **
     pop          -0.03910    0.01557  -2.512  0.01709 * 
     wind         -3.18212    1.85407  -1.716  0.09548 . 
     prep          0.48142    0.35518   1.355  0.18448   
     days         -0.03163    0.15711  -0.201  0.84170   
     ---
     Signif. codes:  0 `***' 0.001 `**' 0.01 `*' 0.05 `.' 0.1 ` ' 1 
     
     Residual standard error: 14.84 on 33 degrees of freedom
     Multiple R-Squared: 0.5304,	Adjusted R-squared: 0.4451 
     F-statistic: 6.213 on 6 and 33 DF,  p-value: 0.0001929 


\end{verbatim}
\begin{enumerate}
\item{}
The multiple regression model for this example can be written as:
\begin{displaymath}
\mbox{\sf SO2}=\beta_0+\beta_1 \, \mbox{\sf temp} +\beta_2 \, \mbox{\sf manu} +\beta_3 \, \mbox{\sf pop} 
+\beta_4 \, \mbox{\sf wind} +\beta_5 \, \mbox{\sf prep} +\beta_6 \, \mbox{\sf days}+\epsilon
\end{displaymath}
where $\epsilon$ represents the error term.

\begin{enumerate}
\item{}
What three assumptions are made about the error terms in a standard regression model?
\item{}
The estimated coefficients  in the output given above were obtained using ``least squares'' estimation.
Briefly explain what this means.
\end{enumerate}
\marks{5}

\newpage

\item{}
Consider the line for {\sf manu} from the R output.
\begin{enumerate}
\item{}
What does the estimated coefficient for {\sf manu} indicate about the effect  {\sf manu}
has on the predicted value of {\sf SO2}? Be precise in your explanation.
\item{}
Explain how you would create a 95\% confidence interval for the 
coefficient of {\sf manu}? Note that you cannot actually produce this interval 
-- just explain how you would do it.
\item{}
What hypothesis is being tested by the P-value (0.00114) on this line
(be precise)?
\end{enumerate}
\marks{5}

\vspace{25mm}

\item
A  plot of studentised residuals versus fitted values for the model 
from part(a) is given below:

\centerline{\resizebox{0.75\textwidth}{0.75\textheight}{\includegraphics{usairp1.pdf}}}

\vspace{5mm}

\begin{enumerate}
\item{}
The studentised residuals are obtained by dividing the ordinary 
residuals by the square root of one minus the hat matrix diagonals: 
$r_i^*=r_i/\sqrt{1-h_{ii}}$.
What advantage do the studentised residuals have over the ordinary residuals?
\item{}
What model deficiency is clearly indicated by this plot? Make sure that 
you explicitly state what pattern in the plot indicates that this problem 
is present.
\item{}
To correct the problem identified in (ii), it was decided 
that $\log (\mbox{\sf SO2})$ should be used as the response. 
Name the three assumptions of the 
regression model that are affected by transforming the response --
one of these should be consistent with your answer to (ii).
\end{enumerate}
\marks{5}

\newpage

\item 
The R output from {\tt all.poss.regs} (using $\log(\mbox{{\sf SO2}})$ as the response) 
and a plot of Mallow's $C_p$ statistic are given below:
\begin{verbatim}

    rssp sigma2 adjRsq     Cp temp manu pop wind prep days
1 12.198  0.321  0.275 24.673    1    0   0    0    0    0
1 13.066  0.344  0.224 28.989    0    0   0    0    0    1
1 14.987  0.394  0.110 38.546    0    1   0    0    0    0
2 10.640  0.288  0.351 18.925    1    0   0    0    0    1
2 10.661  0.288  0.349 19.030    1    0   0    0    1    0
2 10.887  0.294  0.336 20.150    1    1   0    0    0    0
3  9.049  0.251  0.432 13.011    1    1   0    1    0    0
3  9.073  0.252  0.431 13.129    1    0   0    1    1    0
3  9.430  0.262  0.409 14.906    1    0   0    1    0    1
4  7.183  0.205  0.537  5.729    1    1   0    1    1    0
4  7.746  0.221  0.500  8.531    1    1   0    1    0    1
4  8.078  0.231  0.479 10.182    1    1   1    1    0    0
5  6.644  0.195  0.559  5.047    1    1   1    1    1    0
5  7.093  0.209  0.529  7.279    1    1   1    1    0    1
5  7.170  0.211  0.524  7.666    1    1   0    1    1    1
6  6.634  0.201  0.546  7.000    1    1   1    1    1    1

\end{verbatim}


\centerline{\resizebox{0.75\textwidth}{0.75\textheight}{\includegraphics{usairp2.pdf}}}

\vspace{10mm}

Based on this output,  which model or models would you select? 
Justify your choice(s).
\marks{5}


% \centerline{\resizebox{0.75\textwidth}{0.75\textheight}{\includegraphics{usairp3.pdf}}}




\end{enumerate}
