Albert Abbasi, VP of Operations at Ingleside International Bank, is evaluating the service level provided to walk-in customers.Accordingly, he plans a sample of waiting times for walk-in customers.If the population of waiting times has a mean of 15 minutes and a standard deviation of 4 minutes, the probability that Albert's sample of 64 will have a mean between 13.5 and 16.5 minutes is closest to ________.
A)0.9974
B)0.4987
C)0.9772
D)0.4772
E)0.5000

Question

Quizplus · Accepted Answer

Using the Central Limit Theorem, we can calculate the probability that the sample mean falls within a certain range. The formula for the z-score is $Z = \frac{\bar{X} - \mu}{\sigma/\sqrt{n}}$, where $\bar{X}$ is the sample mean, $\mu$ is the population mean, $\sigma$ is the population standard deviation, and $n$ is the sample size. For the lower limit (13.5 minutes), $Z = \frac{13.5 - 15}{4/\sqrt{64}} = \frac{-1.5}{0.5} = -3$. For the upper limit (16.5 minutes), $Z = \frac{16.5 - 15}{4/\sqrt{64}} = \frac{1.5}{0.5} = 3$. Looking up these Z-scores in the standard normal distribution table, the area between Z = -3 and Z = 3 is approximately 0.9974, indicating the probability.

Albert Abbasi, VP of Operations at Ingleside International Bank, Is