3. A large company is analysing how much money it spends on paper in its offices every year. The number of employees, $x$, and the amount of money spent on paper, $p$ ( $\pounds$ hundreds), in 8 randomly selected offices are given in the table below.

\begin{center}
\begin{tabular}{ | l | c | c | c | c | c | c | c | c | }
\hline
$x$ & 8 & 9 & 12 & 14 & 7 & 3 & 16 & 19 \\
\hline
$p$ (£ hundreds) & 40.5 & 36.1 & 30.4 & 39.4 & 32.6 & 31.1 & 43.4 & 45.7 \\
\hline
\end{tabular}
\end{center}

$$\text { (You may use } \sum x ^ { 2 } = 1160 \quad \sum p = 299.2 \quad \sum p ^ { 2 } = 11422 \quad \sum x p = 3449.5 \text { ) }$$
\begin{enumerate}[label=(\alph*)]
\item Show that $S _ { p p } = 231.92$ and find the value of $S _ { x x }$ and the value of $S _ { x p }$
\item Calculate the product moment correlation coefficient between $x$ and $p$.

The equation of the regression line of $p$ on $x$ is given in the form $p = a + b x$.
\item Show that, to 3 significant figures, $b = 0.824$ and find the value of $a$.
\item Estimate the amount of money spent on paper in an office with 10 employees.
\item Explain the effect each additional employee has on the amount of money spent on paper.

Later the company realised it had made a mistake in adding up its costs, $p$. The true costs were actually half of the values recorded. The product moment correlation coefficient and the equation of the linear regression line are recalculated using this information.
\item Write down the new value of
\begin{enumerate}[label=(\roman*)]
\item the product moment correlation coefficient,
\item the gradient of the regression line.
\end{enumerate}\end{enumerate}

\hfill \mbox{\textit{Edexcel S1 2014 Q3 [16]}}