- A researcher is investigating the distribution of orchids in a field. He believes that the Poisson distribution with a mean of 1.75 may be a good model for the number of orchids in each square metre. He randomly selects 150 non-overlapping areas, each of one square metre, and counts the number of orchids present in each square.
The results are recorded in the table below.
| Number of orchids in | | each square metre |
| 0 | 1 | 2 | 3 | 4 | 5 | 6 |
| Number of squares | 30 | 42 | 35 | 26 | 11 | 6 | 0 |
He calculates the expected frequencies as follows
| Number of orchids in | | each square metre |
| 0 | 1 | 2 | 3 | 4 | 5 | More than 5 |
| Number of squares | 26.07 | 45.62 | 39.91 | 23.28 | 10.19 | 3.57 | \(r\) |
- Find the value of \(r\) giving your answer to 2 decimal places.
The researcher will test, at the \(5 \%\) level of significance, whether or not the data can be modelled by a Poisson distribution with mean 1.75
- State clearly the hypotheses required to test whether or not this Poisson distribution is a suitable model for these data.
The test statistic for this test is 2.0 and the number of degrees of freedom to be used is 4
- Explain fully why there are 4 degrees of freedom.
- Stating your critical value clearly, determine whether or not these data support the researcher's belief.
The researcher works in another field where the number of orchids in each square metre is known to have a Poisson distribution with mean 1.5
He randomly selects 200 non-overlapping areas, each of one square metre, in this second field, and counts the number of orchids present in each square.
- Using a Poisson approximation, show that the probability that he finds at least one square with exactly 6 orchids in it is 0.506 to 3 decimal places.