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6.6 Practice: the central limit theorem

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

Graphing probability

For each problem below:

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

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

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

  • 3.5, 4.25, 0.2441

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

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

  • 0.7499

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

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

  • 4.49 hours

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

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

  • 0.3802

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

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

  • 71.90

Discussion question

What causes the probabilities in [link] and [link] to differ?
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Source:  OpenStax, Collaborative statistics (custom lecture version modified by t. short). OpenStax CNX. Jul 15, 2013 Download for free at http://cnx.org/content/col11543/1.1
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