In a local university, 40% of the students live in the dormitories.A random sample of 80 students is selected for a particular study.The probability that the sample proportion (the proportion living in the dormitories) is between 0.30 and 0.50 is
A) 0.4664
B) 0.9328
C) 0.0336
D) 0.0672

Question

Quizplus · Accepted Answer

We can use the Central Limit Theorem to approximate the proportion distribution as normal distribution with mean = 0.4 and variance = (0.4)(0.6)/80 = 0.00225. Then, we can standardize and calculate the z-scores for the given range:

z1 = (0.30 - 0.4)/sqrt(0.00225) = -4.71
z2 = (0.50 - 0.4)/sqrt(0.00225) = 4.71

Using a standard normal table or a calculator, we can find the area between these two z-scores:

P(-4.71 < Z < 4.71) = 0.9328

Therefore, the probability that the sample proportion is between 0.30 and 0.50 is 0.9328 or choice B.

In a Local University, 40% of the Students Live in the Dormitories.A