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 at least 0.30 is
A) 0.4664
B) 0.9328
C) 0.9664
D) 0.0336

Question

Quizplus · Accepted Answer

The distribution of the sample proportion can be approximated by a normal distribution due to the Central Limit Theorem as the sample size is sufficiently large (greater than 30) and the population is assumed to be normally distributed.

The mean of the sample proportion = population proportion = 0.40 
The standard deviation of the sample proportion = sqrt[(p*(1-p))/n] = sqrt[(0.40*0.60)/80] = 0.055

Now, we need to find the probability that the sample proportion is at least 0.30: 
P(p >= 0.30) = P(z >= (0.30-0.40)/0.055) 
                = P(z >= -1.82) 
                = 0.9664 (using a standard normal distribution table)

Therefore, the best choice is (C) 0.9664.

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