2. The heights, in centimetres, of a random sample of 150 plants of a certain variety were measured. The results are summarised in the histogram.
\includegraphics[max width=\textwidth, alt={}, center]{0e73f1d0-5532-4995-b39e-759d82c2bd92-04_860_1684_367_130} One of the 150 plants is chosen at random, and its height, \(X \mathrm {~cm}\), is noted.

1. Show that \(\mathrm { P } ( 20 < X < 30 ) = 0.147\), correct to 3 significant figures. Sam suggests that the distribution of \(X\) can be well modelled by the distribution \(\mathrm { N } ( 40,100 )\).
2. 1. Give a brief justification for the use of the normal distribution in this context.
   2. Give a brief justification for the choice of the parameter values 40 and 100 .
3. Use Sam's model to find \(\mathrm { P } ( 20 < X < 30 )\). Nina suggests a different model. She uses the midpoints of the classes to calculate estimates, \(m\) and \(s\), for the mean and standard deviation respectively, in centimetres, of the 150 heights. She then uses the distribution \(\mathrm { N } \left( m , s ^ { 2 } \right)\) as her model.
4. Use Nina's model to find \(\mathrm { P } ( 20 < X < 30 )\).
5. 1. Complete the table in the Printed Answer Booklet to show the probabilities obtained from Sam's model and Nina's model.
   2. By considering the different ranges of values of \(X\) given in the table, discuss how well the two models fit the original distribution. Table for (e)(i):

      \(x\)	Below 20	20 to 30	30 to 35	35 to 40	40 to 45	45 to 50	50 to 60	Above 60
      Probability obtained from histogram	0.027	0.147	0.153	0.187	0.193	0.147	0.133	0.013
      Probability obtained from Sam's model, N(40, 100)	0.023		0.150	0.191			0.136	0.023
      Probability obtained from Nina's model, \(\mathrm { N } \left( m , s ^ { 2 } \right)\)	0.030		0.153	0.188			0.130	0.023