Question

Consider a population of size 4 who can be labelled as categories “A”, “A”, “B”, “C”, respectively. Let pA denote the sample proportion that fall into category “A” for a sample of size 3. Find the sampling distribution of pA

Answer 1

N = 4
Total Polpulation = “A”, “A”, “B”, “C”
n= 3
number of unique samples with size 3 = 3! * 2! * 1! = 12
types of Samples will of like : AAB, ABC, BCA, CAA ........

pA denote the sample proportion that fall into category “A” for a sample of size 3
As population size is only 4,with 2 categories belonging to A, there always be at-least one "A" category in any sample of 3.

Also, probability of having two "A" in any sample of size 3 = 6/12 = 0.5

So sampling distribution will be: