Question

Suppose that 29% of all residents of a community favor annexation by a nearby municipality. Find the probability that in a random sample of 50 residents, at least 35% will favor annexation. First, verify that the sample is sufficiently large to use the normal distribution

Answer 1

Let X be a random variable denoting the number of residents in the sample who favor annexation.

X follows a binomial distribution with parameters n=50 and p=0.29

Mean of X = n*p = 50*0.29 = 14.5

Standard deviation of X = sqrt(n*p*(1-p)) = 3.2086

35% of 50 =17.5

Normal approximation can be applied as sample size is greater than 31

So, Required Probability = P(X>=17.5) = 1 - P(X<17.5) = 1 - P(z<(17.5-14.5)/3.2086) = 1 - P(z<0.9350) = 1- 0.825106 = 0.174894