In an ecological study of the feeding behaviour of birds, the number of hops between flights

was counted for several birds and the data are given below.

A scientist proposed to model this data using a geometric distribution of the form given below

with parameter $p,$ where $p$ is the probability of a hop resulting in a successful flight.

$f(y)=p(1-p{)}^{y-1}$;$y=1,2,3,$dots.

a$)$ Fit this geometric distribution to the data, by calculating the likelihood

function, and obtain the MLE of the parameter p$.$ State the assumptions

required in constructing the likelihood. Find an approximate $95\%$ confidence

interval for the parameter $p.$

b$)$ Perform a test to examine the goodness of fit of the model and give your

conclusions.

c$)$ On another occasion, the scientist wants to model the number of hops

between flights when more than two hops have taken place and records the

following data.

Discuss how you would modify the likelihood function and obtain a revised

likelihood function for this data. Find the MLE of the parameter $p$ and its

standard error. Based on your results, would you say that a hop has a $40\%$

chance of resulting in a successful flight? Give reasons for your answer.