3 A supermarket sells pears in packs of 8 . Some of the pears in a pack may not be ripe, and the supermarket manager claims that the number of unripe pears in a pack can be modelled by the distribution \(\mathrm { B } ( 8,0.15 )\).
A random sample of 150 packs was selected and the number of unripe pears in each pack was recorded. The following table shows the observed frequencies together with some of the expected frequencies using the manager's binomial distribution.
| Number of unripe pears per pack | 0 | 1 | 2 | 3 | 4 | 5 | \(\geqslant 6\) |
| Observed frequency | 35 | 48 | 43 | 15 | 6 | 3 | 0 |
| Expected frequency | 40.874 | \(p\) | 35.641 | 12.579 | 2.775 | 0.392 | \(q\) |
- Find the values of \(p\) and \(q\).
- Carry out a goodness of fit test, at the \(5 \%\) significance level, to test whether the manager's claim is justified.