6 A bin contains a very large number of paper clips of different colours. The proportion of each colour is shown in the table.
| Colour | White | Yellow | Green | Blue | Red | Purple |
| Proportion | 0.15 | 0.15 | 0.20 | 0.15 | 0.22 | 0.13 |
- Packets are filled from the bin. Each filled packet contains exactly 30 paper clips which may be considered to be a random sample.
Use binomial distributions to determine the probability that a filled packet contains:
- exactly 2 purple paper clips;
- a total of more than 10 red or purple paper clips;
- at least 5 but at most 10 green paper clips.
- Jumbo packets are also filled from the bin. Each filled jumbo packet contains exactly 100 paper clips.
- Assuming that the number of paper clips in a jumbo packet may be considered to be a random sample, calculate the mean and the variance of the number of red paper clips in a filled jumbo packet.
- It is claimed that the proportion of red paper clips in the bin is greater than 0.22 and that jumbo packets do not contain random samples of paper clips.
An analysis of the number of red paper clips in each of a random sample of 50 filled jumbo packets resulted in a mean of 22.1 and a standard deviation of 4.17.
Comment, with numerical justification, on each of the two claims.