A university administrator expects that 25% of students in a core course will receive an A. He looks at the grades assigned to 60 students. The probability that the proportion of students who receive an A is between 0.20 and 0.35 is
A) 0.1867.
B) 0.7776.
C) 0.8133.
D) 0.9633.

Question

Quizplus · Accepted Answer

This problem can be solved using the Central Limit Theorem (CLT) for proportions. The CLT states that if np and n(1-p) are both greater than 5, the sampling distribution of the sample proportion will be approximately normally distributed. Here, n = 60 (number of students), and p = 0.25 (expected proportion of students receiving an A). First, we calculate the mean (μ) and standard deviation (σ) of the sampling distribution of the sample proportion:- μ = p = 0.25- σ = sqrt[(p(1-p))/n] = sqrt[(0.25*0.75)/60] ≈ 0.0566Next, we convert the proportions 0.20 and 0.35 to z-scores:- z for 0.20 = (0.20 - 0.25) / 0.0566 ≈ -0.883- z for 0.35 = (0.35 - 0.25) / 0.0566 ≈ 1.767Using standard normal distribution tables or a calculator, we find the probabilities corresponding to these z-scores:- P(Z < -0.883) ≈ 0.1885- P(Z < 1.767) ≈ 0.9611The probability that the proportion of students who receive an A is between 0.20 and 0.35 is the difference between these two probabilities:- P(0.20 < p < 0.35) = P(Z < 1.767) - P(Z < -0.883) ≈ 0.9611 - 0.1885 = 0.7726The answer closest to this calculated probability is 0.7776, which is option B.

A University Administrator Expects That 25% of Students in a Core