3 The masses, in kilograms, of a random sample of 100 chickens on sale in a large supermarket were recorded as follows.
| Mass \(( m \mathrm {~kg} )\) | \(m < 1.6\) | \(1.6 \leqslant m < 1.8\) | \(1.8 \leqslant m < 2.0\) | \(2.0 \leqslant m < 2.2\) | \(2.2 \leqslant m < 2.4\) | \(2.4 \leqslant m < 2.6\) | \(2.6 \leqslant m\) |
| Frequency | 2 | 8 | 30 | 42 | 11 | 5 | 2 |
- Assuming that the first and last classes are the same width as the other classes, calculate an estimate of the sample mean and show that the corresponding estimate of the sample standard deviation is 0.2227 kg .
A Normal distribution using the mean and standard deviation found in part (i) is to be fitted to these data. The expected frequencies for the classes are as follows.
| Mass \(( m \mathrm {~kg} )\) | \(m < 1.6\) | \(1.6 \leqslant m < 1.8\) | \(1.8 \leqslant m < 2.0\) | \(2.0 \leqslant m < 2.2\) | \(2.2 \leqslant m < 2.4\) | \(2.4 \leqslant m < 2.6\) | \(2.6 \leqslant m\) |
| 2.17 | 10.92 | \(f\) | 33.85 | 19.22 | 5.13 | 0.68 |
- Use the Normal distribution to find \(f\).
- Carry out a goodness of fit test of this Normal model using a significance level of 5\%.
- Discuss the outcome of the test with reference to the contributions to the test statistic and to the possibility of other significance levels.