3The Poisson distribution is also useful for examining the distribution of event

Question

3The Poisson distribution is also useful for examining the distribution of events in space. Suppose you are an ecologist studying stribution of plants. If plants are distributed at random in the landscape, you would expect the number of plants per area sampled to follow a Poisson distribution. However, if the number of plants per area sampled is significantly different from a Poisson distribution, it would provide evidence of interesting, biological processes. For example, if plants were spread out more than expected, it may suggest competition between plants. On the other hand, if plants tended to cluster together, it might suggest other interactions (e.g., adult plants facilitating the growth of seeds). For the Poisson distribution, the mean is equal to the variance. If our sample comes from a Poisson distribution, then the coefficient of dispersion (CD): Should be close to 1 because our sample variance (s) should be similar to our sample mean (2) If s2 »»X, then CD »>1, and samples will show a clustered or clumped distribution. If

Explanation / Answer

For species A, the number of plants in each 36 fields (sampling units) are

3,3,2,2,2,3

1,1,2,3,1,1

2,1,4,2,2,1

3,2,2,0,3,3

1,2,2,2,2,2

2,2,2,1,2,2

Mean of all 36 samples = 1.972

Variance of all 36 samples = 0.6563

CD = Variance / Mean = 0.6563 / 1.972 = 0.3328

The observed CD is less than the critical value of 0.583. So, Species A follows uniform or "spread-out" distribution.

For species B, the number of plants in each 36 fields (sampling units) are

0,0,2,7,6,0
0,7,2,1,2,1
1,5,1,1,1,0
0,1,4,10,7,1
1,0,5,2,2,0
0,2,1,1,6,1

Mean of all 36 samples = 2.25

Variance of all 36 samples = 6.821

CD = Variance / Mean = 6.821 / 2.25 = 3.0316

The observed CD is greater than the critical value of 1.53. So, Species B follows clustered or "clumped" distribution.

For species C, the number of plants in each 36 fields (sampling units) are

2,3,0,1,4,3
1,2,2,1,2,1
2,2,2,3,6,5
2,1,1,2,3,2
3,2,2,3,0,4
3,0,3,7,3,4

Mean of all 36 samples = 2.4167

Variance of all 36 samples = 2.3643

CD = Variance / Mean = 2.3643 / 2.4167 = 0.9783

The observed CD is between the critical values (0.583, 1.53). So, Species C follows Poisson distribution.

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