A random sample of size 1,000 is taken from a population where p = .20.Find P (

Question

A random sample of size 1,000 is taken from a population where p = .20.Find P (  < .22).
A).2643 
B).9429 
C).9207 
D).0571

Quizplus · Accepted Answer

We have a sample size of 1000, and the population proportion is p = 0.20. The sample proportion follows a normal distribution with mean p = 0.20 and standard deviation

standard deviation = sqrt{ [p(1-p)] / n } 
= sqrt{ [0.20(0.80)] / 1000 } 
= 0.01398

We want to find P( p_hat < 0.22 ), which can be rewritten as follows using standardizing the sample proportion:

z = ( p_hat - p ) / standard deviation 
= ( 0.22 - 0.20 ) / 0.01398 
= 1.43

Using a standard normal table or calculator, we can find P( z < 1.43 ) = 0.9236. Hence, the answer is B) 0.9429.

A Random Sample of Size 1,000 Is Taken from a Population