Estimate percentages or proportions from graphs

1. \begin{figure}[h] \end{figure} The histogram in Figure 1 shows the times taken to complete a crossword by a random sample of students. The number of students who completed the crossword in more than 15 minutes is 78
Estimate the percentage of students who took less than 11 minutes to complete the crossword.

1. 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.

1. 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 \(\mathrm { N } \left( 14.9,9.3 ^ { 2 } \right)\)
2. 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
   $$\int _ { 0 } ^ { n } x \mathrm { e } ^ { - x } \mathrm {~d} x = 1 - ( n + 1 ) \mathrm { e } ^ { - n }$$
3. Show that, for Xiang's model, \(k = 99\) to the nearest integer.
4. Estimate \(\mathrm { P } ( 10 < T < 30 )\) using
   1. Tomas' model of \(T \sim \mathrm {~N} \left( 14.9,9.3 ^ { 2 } \right)\)
   2. 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 \(\mathrm { P } ( a < T < b )\)
5. State one limitation of Xiang's model. \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Question 6 continued} \includegraphics[alt={},max width=\textwidth]{a067577e-e2a6-440b-9d22-d558fade15f0-17_1164_1778_294_146}
   \end{figure} Time in hours

2. The box plot in Figure 1 shows a summary of the weights of the luggage, in kg, for each musician in an orchestra on an overseas tour. \begin{figure}[h] \end{figure} The airline's recommended weight limit for each musician's luggage was 45 kg . Given that none of the musicians' luggage weighed exactly 45 kg ,

1. state the proportion of the musicians whose luggage was below the recommended weight limit. A quarter of the musicians had to pay a charge for taking heavy luggage.
2. State the smallest weight for which the charge was made.
3. Explain what you understand by the + on the box plot in Figure 1, and suggest an instrument that the owner of this luggage might play.
4. Describe the skewness of this distribution. Give a reason for your answer. One musician of the orchestra suggests that the weights of luggage, in kg, can be modelled by a normal distribution with quartiles as given in Figure 1.
5. Find the standard deviation of this normal distribution.