Question

Suppose that 29% of all residents of a community favor annexation by a nearby municipality. The probability that in a random sample of 50 residents at least 35% will favor annexation is

Answer 1

Let us say that,

X = the number of residents in the sample who favor annexation.
X has a distribution which follows a binomial curve with parameters:

n=50 and p=0.29

Calculating for mean:
Mean of X = n * p = 50 * 0.29

Mean of X = 14.5

Calculating for standard deviation:
Standard deviation of X = sqrt(n * p * (1 - p))

Standard deviation of X = 3.2086

Now we are to find the probability that at least 35% favour annexation:
35% * 50 = 17.5 residents

Normal approximation can be applied in this case since sample size is greater than 31. Therefore,

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

Answer:

0.175 or 17.5%