Each member of a group of 27 people was timed when completing a puzzle.

The time taken, $x$ minutes, for each member of the group was recorded.

These times are summarised in the following box and whisker plot.

\includegraphics{figure_4}

\begin{enumerate}[label=(\alph*)]
\item Find the range of the times.
[1]

\item Find the interquartile range of the times.
[1]

\item For these 27 people $\sum x = 607.5$ and $\sum x^2 = 17623.25$
calculate the mean time taken to complete the puzzle.
[1]

\item calculate the standard deviation of the times taken to complete the puzzle.
[2]

\item Taruni defines an outlier as a value more than 3 standard deviations above the mean.
State how many outliers Taruni would say there are in these data, giving a reason for your answer.
[1]

\item Adam and Beth also completed the puzzle in $a$ minutes and $b$ minutes respectively, where $a > b$.
When their times are included with the data of the other 27 people
\begin{itemize}
\item the median time increases
\item the mean time does not change
\end{itemize}
Suggest a possible value for $a$ and a possible value for $b$, explaining how your values satisfy the above conditions.
[3]

\item Without carrying out any further calculations, explain why the standard deviation of all 29 times will be lower than your answer to part (d).
[1]
\end{enumerate}

\hfill \mbox{\textit{SPS SPS SM 2021 Q4 [10]}}