7.5 Practice: the central limit theorem

Student learning outcomes

• The student will calculate probabilities using the Central Limit Theorem.

Given

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 4 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 $\overline{X}$ be the random variable representing the mean time to complete the 16 reviews. Let $\mathrm{\Sigma X}$ be the total time it takes Yoonie to complete all of the month’s reviews. Assume that the 16 reviews represent a random set of reviews.

Distribution

Complete the distributions.

1. $X$ ~
2. $\overline{X}$ ~
3. $\mathrm{\Sigma X}$ ~

Graphing probability

For each problem below:

• a Sketch the graph. Label and scale the horizontal axis. Shade the region corresponding to the probability.
• b Calculate the value.

Discussion question