# 7.5 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 $X$ be the random variable representing the time it takes her to complete one review. Assume $X$ is normally distributed. Let $\overline{X}$ be the random variable representing the mean time to complete the 16 reviews. Let $\mathrm{\Sigma X}$ 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. $\overline{X}$ ~
3. $\mathrm{\Sigma X}$ ~

## 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.

• $P\left($ ________ $\begin{array}{} ________ $\right)=$ _______

• 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.

• $P\text{(________________ )}=$ _______

• 0.7499

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

• The 95th Percentile=

• 4.49 hours

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

• The Probability=

• 0.3802

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

• The 95th percentile=

• 71.90

## Discussion question

What causes the probabilities in [link] and [link] to differ?

#### Questions & Answers

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An investment account was opened with an initial deposit of $9,600 and earns 7.4% interest, compounded continuously. How much will the account be worth after 15 years? Kala Reply lim x to infinity e^1-e^-1/log(1+x) given eccentricity and a point find the equiation Moses Reply A soccer field is a rectangle 130 meters wide and 110 meters long. The coach asks players to run from one corner to the other corner diagonally across. What is that distance, to the nearest tenths place. Kimberly Reply Jeannette has$5 and \$10 bills in her wallet. The number of fives is three more than six times the number of tens. Let t represent the number of tens. Write an expression for the number of fives.
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. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
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1 It is estimated that 30% of all drivers have some kind of medical aid in South Africa. What is the probability that in a sample of 10 drivers: 3.1.1 Exactly 4 will have a medical aid. (8) 3.1.2 At least 2 will have a medical aid. (8) 3.1.3 More than 9 will have a medical aid.
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Source:  OpenStax, Collaborative statistics. OpenStax CNX. Jul 03, 2012 Download for free at http://cnx.org/content/col10522/1.40
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