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 between 0.30 and 0.50 is
A)0.4664
B)0.9328
C)0.0336
D)0.0672

Question

Quizplus · Accepted Answer

To find the probability that the sample proportion is between 0.30 and 0.50, we first need to find the mean and standard deviation of the sampling distribution of the sample proportion. The mean of the sampling distribution of the sample proportion is equal to the population proportion, which is 0.40. The standard deviation of the sampling distribution can be found using the formula:

sqrt[(p*(1-p))/n] = sqrt[(0.4*(1-0.4))/80] = 0.0511

Next, we need to standardize the interval (0.30, 0.50) using the formula:

(z - mean)/standard deviation

where z is the standardized value, mean is the population proportion of 0.40, and standard deviation is 0.0511.

For the lower limit of 0.30:

(z - 0.40)/0.0511 = -1.96

z = -0.104

For the upper limit of 0.50:

(z - 0.40)/0.0511 = 1.96

z = 0.896

Therefore, the probability that the sample proportion is between 0.30 and 0.50 is the area under the standard normal curve between z = -0.104 and z = 0.896, which is 0.9328.

Thus, the best choice is B.

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