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
Note that for 5 gaps, the probability and expected count that
is calculated assumes that category to represent ≥ 5 gaps, NOT just 5.
P(X≥5) = 1-(0.135+0.271+0.271+0.180+0.090+0.053)
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)?
χ20.10,4=7.779;
4 df because we subtracted an extra df for the estimation of the mean, α = 0.
10
Fail to Reject H0: We cannot say that
the number of gaps follows a distribution different from a Poisson with estimated
mean = 2.0