Question

QUESTION

The table below shows data collected over a two-hundred-year time span in a woodland area. Each year, the number of new invasive species that appeared for the very first time in that woodland was recorded. Do new invasive species appear in random years, or non-random years? If they appear randomly, the data should follow a Poisson distribution.

1. Calculate the mean number of new invasive species per year
2. Test whether the data fit a Poisson distribution, making sure the data meet the assumptions of a Chi-square goodness-of-fit test.

Number of new invasive species arriving in a year (these are the categories)	Number of years (number of observed years, out of 200, in which that number of new invasive spp. occurred)
0	109
1	65
2	22
3	3
4	1
>4	0
	Total: 200 yrs

Answer 1

a)

X	observed (O)	X*O
0	109	0
1	65	65
2	22	44
3	3	9
4	1	4

total observation = ΣO =200      
expected frequency = ΣX*O/n = 122/200=0.61      
µ = 122/200 = 0.61

b)

expected proportion = e-µ * µ^x / x!      
expected frequency = expected proportion * total observation      

X	observed (O)	X*O	expected proportion = e-µ * µ^x / x!	expected frequency ( E )	(O-E)²/E
0	109	0	0.543	108.670	0.001
1	65	65	0.331	66.289	0.025
2	22	44	0.101	20.218	0.157
3	3	9	0.021	4.111	0.300
4	1	4	0.003	0.627	0.222

chi square test statistic,X² = Σ(O-E)²/E =   0.71              
                  
level of significance, α=   0.05              
Degree of freedom=k-2 =    5   -   2   =   3
                  
P value =   0.8719   [ excel function: =chisq.dist.rt(test-stat,df) ]          
Decision: P value >α , Do not reject Ho          

Hence not good fit for Poission distribution.
                  

Please revert back in case of any doubt.

Please upvote. Thanks in advance.