- A medical researcher is studying the number of hours, \(T\), a patient stays in hospital following a particular operation.
The histogram on the page opposite summarises the results for a random sample of 90 patients.
- Use the histogram to estimate \(\mathrm { P } ( 10 < T < 30 )\)
For these 90 patients the time spent in hospital following the operation had
- a mean of 14.9 hours
- a standard deviation of 9.3 hours
Tomas suggests that \(T\) can be modelled by \(\mathrm { N } \left( 14.9,9.3 ^ { 2 } \right)\) - With reference to the histogram, state, giving a reason, whether or not Tomas' model could be suitable.
Xiang suggests that the frequency polygon based on this histogram could be modelled by a curve with equation
$$y = k x \mathrm { e } ^ { - x } \quad 0 \leqslant x \leqslant 4$$
where
- \(x\) is measured in tens of hours
- \(k\) is a constant
- Use algebraic integration to show that
$$\int _ { 0 } ^ { n } x \mathrm { e } ^ { - x } \mathrm {~d} x = 1 - ( n + 1 ) \mathrm { e } ^ { - n }$$ - Show that, for Xiang's model, \(k = 99\) to the nearest integer.
- Estimate \(\mathrm { P } ( 10 < T < 30 )\) using
- Tomas' model of \(T \sim \mathrm {~N} \left( 14.9,9.3 ^ { 2 } \right)\)
- Xiang's curve with equation \(y = 99 x \mathrm { e } ^ { - x }\) and the answer to part (c)
The researcher decides to use Xiang's curve to model \(\mathrm { P } ( a < T < b )\)
- State one limitation of Xiang's model.
\begin{figure}[h]
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\caption{Question 6 continued}
\includegraphics[alt={},max width=\textwidth]{a067577e-e2a6-440b-9d22-d558fade15f0-17_1164_1778_294_146}
\end{figure}
Time in hours