A random sample of size 100 is taken from a population described by the proportion p = 0.60. The probability that the sample proportion is between 0.55 and 0.62 is ________.
A) 0.1539
B) 0.5047
C) 0.6591
D) 0.8130

Question

Quizplus · Accepted Answer

We can use the Central Limit Theorem (CLT) to approximate the distribution of the sample proportion, which will follow a normal distribution with mean = p = 0.60 and standard deviation = sqrt[(p(1-p))/n] = sqrt[(0.60*0.40)/100] = 0.04899.

To find the probability that the sample proportion is between 0.55 and 0.62, we need to standardize the values using the formula z = (x - p)/sqrt[(p(1-p))/n].

For x = 0.55, z = (0.55 - 0.60)/0.04899 = -1.02
For x = 0.62, z = (0.62 - 0.60)/0.04899 = 0.41

Using a standard normal distribution table or calculator, the probability of z being between -1.02 and 0.41 is approximately 0.5047. Therefore, the answer is B.

A Random Sample of Size 100 Is Taken from a Population