The Office of Student Services at a large western state university maintains information on the study habits of its full-time students. Their studies indicate that the mean amount of time undergraduate students study per week is 20 hours. The hours studied follows the normal distribution with a population standard deviation of six hours. Suppose we select a random sample of 144 current students. What is the probability that the mean of this sample is between 19 hours and 20 hours?
A)2.00
B)0.4772
C)−2.00
D)Cannot be calculated based on the given information.

Question

Quizplus · Accepted Answer

We are given that the population mean is μ = 20, population standard deviation is σ = 6, sample size is n = 144, and we need to find the probability that the sample mean falls between 19 and 20 hours, or P(19 ≤ X̄ ≤ 20).
We can use the central limit theorem to convert the distribution of sample means to a standard normal distribution:
z = (X̄ - μ) / (σ / √n)
z1 = (19 - 20) / (6 / √144) = -2
z2 = (20 - 20) / (6 / √144) = 0
P(19 ≤ X̄ ≤ 20) = P(-2 ≤ z ≤ 0)
Using a standard normal table or calculator, we find this probability to be approximately 0.4772. Therefore, the best answer is B.

The Office of Student Services at a Large Western State