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\title{Correlation and Regression\footnote{\copyright 2019 Joseph C Watkins}}
\author{Worksheet 3}
\date{}
\maketitle

\begin{enumerate}
\item For a given day, the number of heating degrees is 65 minus the mean temperature in degrees Fahrenheit. If the mean is above 65, the  number of heating degrees is 0 for that day. {\em Heating degree days} is the sum of the heating degrees over a certain period (say a month). They are commonly used in calculations relating to the energy consumption required to heat buildings. We will use linear regression to predict kilowatt hour (kWh) energy use from heating degree days (hdd) (for a particular house in Scotland). Here are the data. \index{heating degree days}
\begin{small}
\begin{center}
\begin{tabular}{c|cccccccccccc}
Month&Oct&Nov&Dec&Jan&Feb&Mar&Apr&May&Jun&Jul&Aug&Sep \\ \hline
hdd&163&228&343&373&301&238&137&84&38&15&14&34 \\
kWh&593&676&1335&1149&1127&892&538&289&172&131&134&134
\end{tabular}
\end{center}
\end{small}
\begin{enumerate}
\item Which is the explanatory variable and which is the response?
\item Display a scatterplot and describe any structure you see in the data.
\item Give the equation of the regression line and place it on a scatterplot.
\item What is the predicted energy use for a month in which the average temperature is $50^\circ$F and no day has an average temperature above $65^\circ$F? 
\item In a cooler climate, the baseload energy or non-weather-dependent consumption is the amount of energy used when none is devoted to heating. Estimate this using the regression line.
\item Find the correlation of these two quantitative variables. What does this value for the correlation tell you?
\end{enumerate}
\item Global warming has many indirect effects on climate. For example, summer monsoon winds in the Arabian Sea bring rain to India needed for agriculture. As the climate warms and winter snow cover in Europe and Asia decreases, the land heats up more rapidly in summer which may increase the strength of the monsoon. The data are snow cover (in millions of square kilometers) and stress (in newtons per square meter). \index{global warming}
\begin{verbatim} 
> snowcover<-c(6.6,5.9,6.8,7.7,7.9,7.8,8.1,16.6,18.2,15.2,16.2,17.1,17.3,
+ 18.1,26.6,27.1,27.5,28.4,28.6,29.6,29.4)
> windstress<-c(0.125,0.160,0.158,0.155,0.169,0.173,0.196,0.111,0.106,0.
+ 143,0.153,0.155,0.133,0.130,0.062,0.051,0.068,0.055,0.033,0.029,0.024)
\end{verbatim}
\begin{enumerate}
\item Give the correlation between snow cover and wind stress.
\item Give a scatterplot with snow cover as the explanatory variable.
\item Find the equation for the corresponding regression line and add it to the scatterplot.
\item Use this to predict wind stress when snow cover is 12 million square kilometers.
\item Give the regression line using wind stress as the explanatory variable and add it to the scatterplot.
\item Take the predicted value for wind stress in part (d) and use this to predict snow cover. Is the value 12 million square kilometers?
\item What does you findings in part (f) tell you?
\end{enumerate}
\end{enumerate}
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