An ecologist studied the distribution of the moss species Hypnum schreberi invading mica residues of clay deposits. She laid out 126 quadrats and counted the number of moss shoots in each. Let X be the number of shoots observed in a quadrat and let f denote the observed number of quadrats containing a given number of shoots. The ecologist obtained the following results:

X	0	1	2	3	4	5	>5
f	100	9	8	6	2	1	0

a) Assuming X follows a Poisson distribution, determine a 95% confidence interval for the mean number of moss shoots in a quadrat.

b) Calculate the variance mean ratio.

c) Calculate the Poisson index of dispersion.

d) Use the index calculated above to test whether the distribution of moss shoots follows Poisson distribution. Use 5% level of significance.

e) Use chi-square goodness of fit test to test whether the distribution of moss shoots follows Poisson distribution. Use 5% level of significance.