- The manager of a supermarket is investigating the number of complaints per day received from customers.
A random sample of 180 days is taken and the results are shown in the table below.
| Number of complaints per day | 0 | 1 | 2 | 3 | 4 | 5 | 6 | \(\geqslant 7\) |
| Frequency | 12 | 28 | 37 | 38 | 29 | 17 | 19 | 0 |
- Calculate the mean and the variance of these data.
- Explain why the results in part (a) suggest that a Poisson distribution may be a suitable model for the number of complaints per day.
The manager uses a Poisson distribution with mean 3 to model the number of complaints per day.
- For a randomly selected day find, using the manager's model, the probability that there are
- at least 3 complaints,
- more than 4 complaints but less than 8 complaints.
A week consists of 7 consecutive days.
- Using the manager's model and a suitable approximation, show that the probability that there are less than 19 complaints in a randomly selected week is 0.29 to 2 decimal places.
Show your working clearly.
(Solutions relying on calculator technology are not acceptable.)
A period of 13 weeks is selected at random. - Find the probability that in this period there are exactly 5 weeks that have less than 19 complaints.
Show your working clearly.