Exhibit 7-3
In a local university, 40% of the students live in the dormitories. A random sample of 80 students is selected for a particular study.
-Refer to Exhibit 7-3. 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

We are dealing with a binomial distribution (students living in dorms or not) and the sample size (80) is large enough to use the normal approximation to the binomial distribution. The mean of the sample proportion is p = 0.40 and the standard deviation is calculated as sqrt[(p*(1-p))/n] = sqrt[(0.4*0.6)/80] = 0.06. 
To find the probability that the sample proportion is at least 0.30, we need to find the area to the right of 0.30 under the standard normal distribution. Using z-score formula, z = (0.30-0.40)/0.06 = -1.67. Consulting a standard normal distribution table or using a calculator, we find that the area to the right of -1.67 is 0.9525. However, since we want the area to the right of 0.30 (not to the left of 0.30), we subtract this value from 1: 1 - 0.9525 = 0.0475. Thus, the probability that the sample proportion is at least 0.30 is 0.0475 or 4.75%. Therefore, the best choice is (C).

Exhibit 7-3 in a Local University, 40% of the Students Live in Live