A random sample of size 1,000 is taken from a population where p = .20.Find $\mathrm { P } ( \hat { p }

Question

A random sample of size 1,000 is taken from a population where p = .20.Find $\mathrm { P } ( \hat { p } < .18 )$
A).9429
B).0571
C).2643
D).0793

Quizplus · Accepted Answer

In this case, the population proportion (p) is known and the sample size (n) is large enough (n=1000), so we can use the normal approximation to the binomial distribution. The mean of the sampling distribution of the sample proportion is equal to p=0.20, and the standard deviation is equal to sqrt(p*(1-p)/n) = sqrt(0.20*0.80/1000) = 0.0141. To find P(\hat{p} < 0.18), we standardize the sample proportion as (0.18 - 0.20)/0.0141 = -1.42 and look up the area to the left of -1.42 under the standard normal curve, which is 0.0793. Therefore, the answer is B.

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