Question

Suppose a simple random sample of size n=75 is obtained from a population whose size is N= 30,000 and whose population proportion with a specified characteristic is p= 0.4 .

A) Determine the standard deviation of the sampling distribution of p hat (Round to 6 decimals)

B) What is the probability of obtaining x=33 or more individuals with the​ characteristic? That​ is, what is ​P(p ≥0.44​)? (Round to 4 decimals)

Answer 1

Given sample size n=75, population size N=30000 and population proportion with specified characteristic p=0.4.

(a) assumption is satisfied here. Hence shape,mean and standard deviation of sampling distribution of can be determined.

Now, . hence the distribution of is approximately normal.

Mean of the sampling distribution of is

Standard deviation of the sampling distribution of is

(b) For probability of obtaining x=33 or more individuals with the characteristic  we need a z-score so we can find the area to the right of it.

i.e we have

Converting to standard normal random variable Z,

i.e

From normal distribution table the area to the right of z score 0.71 is equal to area to the left of z score -0.71 i.e we get 0.2389. Therefore the answer is