2 A student is investigating the numbers of sultanas in a particular brand of biscuit. The data in the table show the numbers of sultanas in a random sample of 50 of these biscuits.
| Number of sultanas | 0 | 1 | 2 | 3 | 4 | 5 | \(> 5\) |
| Frequency | 8 | 15 | 12 | 9 | 4 | 2 | 0 |
- Show that the sample mean is 1.84 and calculate the sample variance.
- Explain why these results support a suggestion that a Poisson distribution may be a suitable model for the distribution of the numbers of sultanas in this brand of biscuit.
For the remainder of the question you should assume that a Poisson distribution with mean 1.84 is a suitable model for the distribution of the numbers of sultanas in these biscuits.
- Find the probability of
(A) no sultanas in a biscuit,
(B) at least two sultanas in a biscuit. - Show that the probability that there are at least 10 sultanas in total in a packet containing 5 biscuits is 0.4389 .
- Six packets each containing 5 biscuits are selected at random. Find the probability that exactly 2 of the six packets contain at least 10 sultanas.
- Sixty packets each containing 5 biscuits are selected at random. Use a suitable approximating distribution to find the probability that more than half of the sixty packets contain at least 10 sultanas.