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# Practice: Central Limit Theorem (modified R. Bloom)

Module by: Roberta Bloom. E-mail the authorEdited By: Roberta Bloom

Summary: This module contains practice problems for the Central Limit Theorem from Collaborative Statistics by Susan Dean and Dr. Barbara Illowsky. The problems relating to Central Limit Theorem for averages have not been changed in this module. The problems relating to Central Limit Theorem for sums have been removed from this module.

## 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 ~

## 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.
• 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 average of a month’s reviews will take Yoonie from 3.5 to 4.25 hrs.

• a.
• b. P( )=P( )=_______

#### Solution

• b. 0.7499
• NOTE:. When calculating the standard deviation for the mean using the Central Limit Theorem in this problem, the sample size is n = 16, the number of reviews she completes in a month.

### Exercise 3

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

• a.
• b. The 95th Percentile=

#### Solution

• b. 4.49 hours

## Discussion Question

### Exercise 4

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

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