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

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%