*Use the following information to answer the next six exercises:* Yoonie is a personnel manager in a large corporation. Each month she must review 16 of the employees. From past experience, she has found that the reviews take her approximately four hours each to do with a population standard deviation of 1.2 hours. Let \(X\) be the random variable representing the time it takes her to complete one review. Assume \(X\) is normally distributed. Let \(\bar{X}\) be the random variable representing the mean time to complete the 16 reviews. Assume that the 16 reviews represent a random set of reviews.

Example \(\PageIndex{1}\)

What is the mean, standard deviation, and sample size?

**Answer**

mean = 4 hours; standard deviation = 1.2 hours; sample size = 16

Exercise \(\PageIndex{2}\)

Complete the distributions.

- \(X \sim\) _____(_____,_____)
- \(\bar{X} \sim\) _____(_____,_____)

Example \(\PageIndex{3}\)

Find the probability that **one** review will take Yoonie from 3.5 to 4.25 hours. Sketch the graph, labeling and scaling the horizontal axis. Shade the region corresponding to the probability.

**Figure \(\PageIndex{2}\).**

2. \(P\)(________ \(< x <\) ________) = _______

**Answer**

- Check student's solution.
- 3.5, 4.25, 0.2441

Exercise \(\PageIndex{4}\)

Find the probability that the **mean** of a month’s reviews will take Yoonie from 3.5 to 4.25 hrs. Sketch the graph, labeling and scaling the horizontal axis. Shade the region corresponding to the probability.

**Figure \(\PageIndex{3}\).**

2. \(P\)(________________) = _______

Example \(\PageIndex{5}\)

What causes the probabilities in Exercise and Exercise to be different?

**Answer**

The fact that the two distributions are different accounts for the different probabilities.

Exercise \(\PageIndex{6}\)

Find the 95^{th} percentile for the mean time to complete one month's reviews. Sketch the graph.

**Figure \(\PageIndex{4}\).**

- The 95
^{th} Percentile =____________