A
benthic ecologist is studying the pattern of gap formation in intertidal mussel beds.
Unfortunately, he has been unable to find a previous paper listing the
average number of gaps formed in a coast area of a given size. He assumes that since gaps are formed fairly
infrequently and randomly they will follow a Poisson distribution. He also knows that he can estimate the
parameter for his distribution by finding the average in his sample, but that by estimating this parameter from his data,
rather than using a constant from elsewhere (previous research, foraging
theory, etc.), he will lose a degree of freedom (df). [NOTE:
in HW1 #2, the value for mu, the hypothesized
population mean, is given, so you do not have to subtract a degree of freedom
there. However in this lab problem, since we have to estimate the average first
and then use it in computing probabilities, we “pay” for this bit of
statistical information by subtracting one df.] He has the following data from counting the
number of gaps in each of 60 plots. Help
determine if his data follows a Poisson distribution at the a = 0.10 level of
significance.
0
8 0.135 8.120 -0.120 0.0018
1
13 0.271 16.240 -3.240 0.646
2
20 0.271 16.240 3.760 0.870
3
11 0.180 10.827 0.173 0.0028
4
6 0.090 5.413 0.587 0.0636
5
2 0.053 3.159 -1.159 0.425
State
Ho and Ha biologically and statistically.
H0: #
of gaps follows a Poisson distribution with estimated mean
Ha: #
of gaps follows some other distribution
List test assumptions.
see notes
Calculate
the average number of gaps per plot.
(This is your best estimate of m.)
xbar=2 =(0*8+1*13+2*20+3*11+4*6+5*2)/60 = sum(x*p(x))
Find
the probabilities associated with seeing given numbers of gaps in single
plots. (That is, what’s the probability
of seeing a plot with no gaps, with one gap, with two, etc.?) Recall that
and
that the total probability of all the categories (including the possibility
that there are more than 5 gaps in a plot!) must add up to 1.
Use
the probabilities you found to calculate the expected frequencies, and then
find your overall test statistic.
test statistic: 2.01
What’s
your critical value? What do you
conclude (include the p-value)?
=7.779; 4 df
because we subtracted an extra df for the estimation
of the mean
Fail to Reject H0: We cannot say that
the number of gaps follows a distribution different from a Poisson with
estimated mean = 2.0