A random sample of size 100 is taken from a population described by the proportion . Refer to Exhibit 7-6.The probability that the sample proportion is less than 0.55 is ________.
B)0.1539
C)0.3669
D)0.8461

Question

A random sample of size 100 is taken from a population described by the proportion  . Refer to Exhibit 7-6.The probability that the sample proportion is less than 0.55 is ________.
B)0.1539 
C)0.3669 
D)0.8461

Quizplus · Accepted Answer

Since we have a large sample size (n=100) and a known population proportion (p=0.70), we can use the Central Limit Theorem to approximate the sampling distribution of the sample proportion as normal with mean p and standard deviation sqrt(p(1-p)/n). 
Thus, we have: 
P(p_hat < 0.55) = P(z < (0.55-0.70)/sqrt(0.7(1-0.7)/100)) 
= P(z < -2.25) 
Using a standard normal distribution table or calculator, we find that P(z < -2.25) = 0.0122 
Therefore, the probability that the sample proportion is less than 0.55 is approximately 0.0122, which is closest to option B, 0.1539.

A Random Sample of Size 100 Is Taken from a Population