Question

Twenty percent of registered voters are Whigs, 35% are Free Soilers, 35% are Jacksonian Democrats, and 10% are independent. A focus group of 50 voters is randomly selected from the population of registered voters.

(a) What is the probability that the sample contains 10 Whigs, 18 Free Soilers, 17 Jacksonian Democrats and 5 independents? You may use R or any other computational aid to find the answer.

(b) If Jacksonian Democrats and Free Soilers are lumped together as a single voting bloc, what are the expected value and variance of the number in the sample that belongs to this bloc?

Answer 1

We have given that n=50

p1=0.20,p2=0.35,p3=0.35,p3=0.10

By using Multinomial Distribution.

(a) the probability that the sample contains 10 Whigs, 18 Free Soilers, 17 Jacksonian Democrats and 5 independents is

Or by using R Command,

x=c(10,18,17,5)
prob=c(0.20,0.35,0.35,0.10)
dmultinom(x,size=50,prob)

probability=0.0035

(b) If Jacksonian Democrats and Free Soilers are lumped together as a single voting bloc then

p=0.70 and n=35

This follows Binomial distribution with p=0.70 and n=35.

We know that the mean and variance of binomial distribution are

The expected value=mean=n*p=35*0.70=24.5

Variance=n*p*(1-p)=35*0.70*0.30=7.35