11 As part of a research project, the masses, \(m\) grams, of a random sample of 1000 pebbles from a certain beach were recorded. The results are summarised in the table.
| Mass \(( \mathrm { g } )\) | \(50 \leqslant m < 150\) | \(150 \leqslant m < 200\) | \(200 \leqslant m < 250\) | \(250 \leqslant m < 350\) |
| Frequency | 162 | 318 | 355 | 165 |
- Calculate estimates of the mean and standard deviation of these masses.
The masses, \(x\) grams, of a random sample of 1000 pebbles on a different beach were also found. It was proposed that the distribution of these masses should be modelled by the random variable \(X \sim \mathrm {~N} ( 200,3600 )\).
- Use the model to find \(\mathrm { P } ( 150 < X < 210 )\).
- Use the model to determine \(x _ { 1 }\) such that \(\mathrm { P } \left( 160 < X < x _ { 1 } \right) = 0.6\), giving your answer correct to five significant figures.
It was found that the smallest and largest masses of the pebbles in this second sample were 112 g and 288 g respectively.
- Use these results to show that the model may not be appropriate.
- Suggest a different value of a parameter of the model in the light of these results.