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 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 less than 19.25 hours?
A) 0.4332
B) 0.8664
C) 0.0668
D) 0.0181

Question

Quizplus · Accepted Answer

We need to use the Central Limit Theorem, which allows us to assume that the distribution of sample means is approximately normal, with a mean of 20 and a standard deviation of 6 / sqrt(144) = 0.5. We can standardize the sample mean as follows:

z = (x̄ - μ) / (σ / sqrt(n))
z = (19.25 - 20) / (0.5)
z = -1.5

Then, we need to find the probability of getting a z-score less than -1.5. Using a standard normal table or calculator, we can find that this is equivalent to a probability of 0.0668. Therefore, the answer is C.

The Office of Student Services at a Large Western State