6. A total of 100 random samples of 6 items are selected from a production line in a factory and the number of defective items in each sample is recorded. The results are summarised in the table below.
- Show that the mean number of defective items per sample is 2.91
A factory manager suggests that the data can be modelled by a binomial distribution with \(n = 6\). He uses the mean from the sample above and calculates expected frequencies as shown in the table below.
| 0 | 1 | 2 | 3 | 4 | 5 | 6 |
| 1.87 | 10.54 | 24.82 | \(a\) | 22.01 | 8.29 | \(b\) |
- Calculate the value of \(a\) and the value of \(b\) giving your answers to 2 decimal places.
- Test, at the \(5 \%\) level, whether or not the binomial distribution is a suitable model for the number of defective items in samples of 6 items. State your hypotheses clearly.