3 A new car tyre is fitted to a wheel. The tyre is inflated to its recommended pressure of 265 kPa and the wheel left unused. At 3-month intervals thereafter, the tyre pressure is measured with the following results:

\begin{center}
\begin{tabular}{ | l | c | c | c | c | c | c | c | c | c | }
\hline
\begin{tabular}{ l }
Time after fitting \\
$( x$ months $)$ \\
\end{tabular} & 0 & 3 & 6 & 9 & 12 & 15 & 18 & 21 & 24 \\
\hline
\begin{tabular}{ l }
Tyre pressure \\
$( y$ kPa $)$ \\
\end{tabular} & 265 & 250 & 240 & 235 & 225 & 215 & 210 & 195 & 180 \\
\hline
\end{tabular}
\end{center}
\begin{enumerate}[label=(\alph*)]
\item \begin{enumerate}[label=(\roman*)]
\item Calculate the equation of the least squares regression line of $y$ on $x$.
\item Interpret in context the value for the gradient of your line.
\item Comment on the value for the intercept with the $y$-axis of your line.
\end{enumerate}\item The tyre manufacturer states that, when one of these new tyres is fitted to the wheel of a car and then inflated to 265 kPa , a suitable regression equation is of the form

$$y = 265 + b x$$

The manufacturer also states that, as the car is used, the tyre pressure will decrease at twice the rate of that found in part (a).
\begin{enumerate}[label=(\roman*)]
\item Suggest a suitable value for $b$.
\item One of these new tyres is fitted to the wheel of a car and inflated to 265 kPa . The car is then used for 8 months, after which the tyre pressure is checked for the first time.

Show that, accepting the manufacturer's statements, the tyre pressure can be expected to have fallen below its minimum safety value of 220 kPa .\\
(2 marks)
\end{enumerate}\end{enumerate}

\hfill \mbox{\textit{AQA S1 2006 Q3 [11]}}