6 A practice examination paper is taken by 500 candidates, and the organiser wishes to know what continuous distribution could be used to model the actual time, \(X\) minutes, taken by candidates to complete the paper.
The organiser starts by carrying out a goodness-of-fit test for the distribution \(\mathrm { N } \left( 100,15 ^ { 2 } \right)\) at the \(5 \%\) significance level. The grouped data and the results of some of the calculations are shown in the following table.
| Time | \(0 \leqslant X < 80\) | \(80 \leqslant X < 90\) | \(90 \leqslant X < 100\) | \(100 \leqslant X < 110\) | \(X \geqslant 110\) |
| Observed frequency \(O\) | 36 | 95 | 137 | 129 | 103 |
| Expected frequency \(E\) | 45.606 | 80.641 | 123.754 | 123.754 | 126.246 |
| \(\frac { ( O - E ) ^ { 2 } } { E }\) | 2.023 | 2.557 | 1.418 | 0.222 | 4.280 |
- State suitable hypotheses for the test.
- Show how the figures 123.754 and 0.222 in the column for \(100 \leqslant X < 110\) were obtained. [3]
- Carry out the test.
The organiser now wants to suggest an improved model for the data.
- Suggest an aspect of the data that the organiser should take into account in considering an improved model.
- The graph of the probability density function for the distribution \(\mathrm { N } \left( 100,15 ^ { 2 } \right)\) is shown in the diagram in the Printed Answer Booklet.
On the same diagram sketch the probability density function of an improved model that takes into account the aspect of the data in part (d)(i).