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Practice: The Central Limit Theorem

Module by: Susan Dean, Barbara Illowsky, Ph.D.. E-mail the authors

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 XX size 12{X} {} be the random variable representing the time it takes her to complete one review. Assume XX size 12{X} {} is normally distributed. Let X¯X¯ size 12{ {overline {X}} } {} be the random variable representing the mean time to complete the 16 reviews. Let ΣXΣX size 12{ΣX} {} 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. XX ~
  2. X¯X ~
  3. ΣXΣX ~

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.

Exercise 1

Find the probability that one review will take Yoonie from 3.5 to 4.25 hours.

  • a. Empty normal distribution curve graph with x-axis of x.
  • b. P(P( size 12{P \( } {} ________ <x<<x< size 12{<x<} {} ________ )=)= size 12{ \) ={}} {} _______

Solution

  • b. 3.5, 4.25, 0.2441

Exercise 2

Find the probability that the mean of a month’s reviews will take Yoonie from 3.5 to 4.25 hrs.

  • a. Empty normal distribution curve graph for the average.
  • b. P(________________ )=P(________________ )=_______

Solution

  • b. 0.7499

Exercise 3

Find the 95th percentile for the mean time to complete one month’s reviews.

  • a. Empty normal distribution curve graph for the mean.
  • b. The 95th Percentile=

Solution

  • b. 4.49 hours

Exercise 4

Find the probability that the sum of the month’s reviews takes Yoonie from 60 to 65 hours.

  • a. Empty normal distribution curve graph for Sum X.
  • b. The Probability=

Solution

  • b. 0.3802

Exercise 5

Find the 95th percentile for the sum of the month’s reviews.

  • a. Empty normal distribution curve graph for Sum X.
  • b. The 95th percentile=

Solution

  • b: 71.90

Discussion Question

Exercise 6

What causes the probabilities in Exercise 1 and Exercise 2 to differ?

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