Question

Suppose you want to estimate a particular population proportion p of “success”, say the proportion of Cal Poly students who plan to go to Coachella this year. Consider two methods of collecting data.

1) Select a simple random sample of size n for a fixed, specified n. Let X be the count of successes in the sample. For example, select a sample of n = 30 students, and say for the selected sample X = 3 students plan to attend Coachella.

2) Select individuals one at a time until x successes are obtained and then stop sampling, for a fixed, specified x. Let N be the total number of individuals selected from the sample. For example, sample students until obtaining a total of x = 3 who plan to attend Coachella, and say for the selected sample N = 30 were needed (the 30th student selected is the 3rd success).

(a) Write the likelihood function L(p) for method 1. What is the MLE of p?

(b) Let θ1 be the probability that the sample proportion equals 0.1 when n = 30 with method 1. Find the MLE of θ1.

(c) Write the likelihood function L(p) for method 2. Without doing any further calculus find the MLE of p based on method 2. Explain your reasoning. (Hint: compare the likelihood functions for method 1 and method 2 as functions of p.)

(d) Let θ1 be the probability that the sample proportion equals 0.1 when x = 3 with method 2. Find the MLE of θ2

Answer 1

The solution should be as follows: