The article “Feeding Ecology of the Red-Eyed Vireo and Associated Foliage-Gleaning Birds” (Ecological Monographs, 1971: 129–152) presents the accompanying data on the variable the number of hops before the first flight and preceded by a flight. The author then proposed and fit a geometric probability distribution for , where to the data.
The total sample size was .
x 1 2 3 4 5 6 7 8 9 10 11 12 Number of Times x 48 31 20 9 6 5 4 2 1 1 2 1 Observed

a. The likelihood is . Show that the mle of p is given by , and compute for the given data.

b. Estimate the expected cell counts using of part (a)
[ for ], and test the fit of the model using a x2 test by combining the counts for , and 12 into one cell .