4 A chemist is conducting an experiment in which the concentration of a certain chemical, A , is supposed to be recorded at the start of the experiment and then every 30 seconds after the start. The time after the start is denoted by $t \mathrm {~s}$ and the concentration by $\mathrm { z } \mathrm { mg } \mathrm { cm } ^ { - 3 }$. The collected data are shown in the table below. Note that the concentration at $t = 90$ was not recorded.

\begin{center}
\begin{tabular}{ | l | c | c | c | c | c | }
\hline
Time, $t$ & 0 & 30 & 60 & 120 & 150 \\
\hline
Concentration of A, $z$ & 40.0 & 31.3 & 27.5 & 12.8 & 11.4 \\
\hline
\end{tabular}
\end{center}

The chemist wishes to plot the data on a graph.
\begin{enumerate}[label=(\alph*)]
\item Explain why $t$ should be plotted on the horizontal axis.

You are given that the summary statistics for the data are as follows.\\
$n = 5 \quad \sum t = 360 \quad \sum z = 123.0 \quad \sum t ^ { 2 } = 41400 \quad \sum z ^ { 2 } = 3629.74 \quad \sum \mathrm { t } = 5835$

The regression line of $z$ on $t$ is given by $\mathbf { z = a + b t }$ and is used to model the concentration of chemical A for $t \geqslant 0$.
\item \begin{enumerate}[label=(\roman*)]
\item Use the summary statistics to determine the value of $a$ and the value of $b$.
\item Find the value of the residual at each of the following values of $t$.

\begin{itemize}
\end{enumerate}\item $t = 60$
  \item $t = 120$
\item \begin{enumerate}[label=(\roman*)]
\item Use the equation of the regression line to estimate the value of the concentration at 90 seconds.
\item With reference to your answers to part (b)(ii), comment on the reliability of your answer to part (c)(i).
\end{itemize}

Further experiments indicate that the model is reasonably reliable for times greater than 150 seconds up to about 200 seconds.
\end{enumerate}\item Show that the model cannot be valid beyond a time of about 200 seconds.
\end{enumerate}

\hfill \mbox{\textit{OCR MEI Further Statistics A AS 2024 Q4 [10]}}