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Central Limit Theorem: Practice

Module by: Dr. Barbara Illowsky, Susan Dean

Student Learning Outcomes

  • The student will explore the properties of data through 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 average 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.

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 1

  • b. 3.5, 4.25, 0.2441

Exercise 2

Find the probability that the average 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 2

  • b. 0.7499

Exercise 3

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

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

Solution 3

  • 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 4

  • 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 5

  • b - 71.90

Discussion Question

Exercise 6

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

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