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# Central Limit Theorem: Practice

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## 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.
• 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( )=_______

• b. 0.7499

### 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

### Exercise 4

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

• a.
• b. The Probability=

• b. 0.3802

### Exercise 5

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

• a.
• b. The 95th percentile=

• b: 71.90

## Discussion Question

### Exercise 6

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

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