# 6.3 Using the central limit theorem -- rrc math 1020  (Page 5/31)

 Page 5 / 31

For part a, you include 150 so P ( X ≥ 150) has normal approximation P ( Y ≥ 149.5) = 0.8641.

0.8641.

For part b, you include 160 so P ( X ≤ 160) has normal appraximation P ( Y ≤ 160.5) = 0.5689.

0.5689

For part c, you exclude 155 so P ( X >155) has normal approximation P ( y >155.5) = 0.6572.

0.6572.

For part d, you exclude 147 so P ( X <147) has normal approximation P ( Y <146.5) = 0.0741.

0.0741

For part e, P ( X = 175) has normal approximation P (174.5< Y <175.5) = 0.0083.

0.0083

Because of calculators and computer software that let you calculate binomial probabilities for large values of n easily, it is not necessary to use the the normal approximation to the binomial distribution, provided that you have access to these technology tools. Most school labs have Microsoft Excel, an example of computer software that calculates binomial probabilities. Many students have access to the TI-83 or 84 series calculators, and they easily calculate probabilities for the binomial distribution. If you type in "binomial probability distribution calculation" in an Internet browser, you can find at least one online calculator for the binomial.

For [link] , the probabilities are calculated using the following binomial distribution: ( n = 300 and p = 0.53). Compare the binomial and normal distribution answers. See Discrete Random Variables for help with calculator instructions for the binomial.

P ( X ≥ 150) = 0.8641

P ( X ≤ 160) = 0.5684

P ( X >155) = 0.6576

P ( X <147) = 0.0742

P ( X = 175) = 0.0083

## Try it

In a city, 46 percent of the population favor the incumbent, Dawn Morgan, for mayor. A simple random sample of 500 is taken. Using the continuity correction factor, find the probability that at least 250 favor Dawn Morgan for mayor.

0.0401

## References

Data from the Wall Street Journal.

“National Health and Nutrition Examination Survey.” Center for Disease Control and Prevention. Available online at http://www.cdc.gov/nchs/nhanes.htm (accessed May 17, 2013).

## Chapter review

The central limit theorem can be used to illustrate the law of large numbers. The law of large numbers states that the larger the sample size you take from a population, the closer the sample mean $\overline{x}$ gets to μ .

Use the following information to answer the next ten exercises: A manufacturer produces 25-pound lifting weights. The lowest actual weight is 24 pounds, and the highest is 26 pounds. Each weight is equally likely so the distribution of weights is uniform. A sample of 100 weights is taken.

1. What is the distribution for the weights of one 25-pound lifting weight? What is the mean and standard deivation?
2. What is the distribution for the mean weight of 100 25-pound lifting weights?
3. Find the probability that the mean actual weight for the 100 weights is less than 24.9.
1. U (24, 26), 25, 0.5774
2. N (25, 0.0577)
3. 0.0416

Draw the graph from [link]

Find the probability that the mean actual weight for the 100 weights is greater than 25.2.

0.0003

Draw the graph from [link]

Find the 90 th percentile for the mean weight for the 100 weights.

25.07

Draw the graph from [link]

1. What is the distribution for the sum of the weights of 100 25-pound lifting weights?
2. Find P ( Σx <2,450).
1. N (2,500, 5.7735)
2. 0

Draw the graph from [link]

Find the 90 th percentile for the total weight of the 100 weights.

2,507.40

Draw the graph from [link]

Use the following information to answer the next five exercises:
The length of time a particular smartphone's battery lasts follows an exponential distribution with a mean of ten months. A sample of 64 of these smartphones is taken.

1. What is the standard deviation?
2. What is the parameter m ?
1. 10
2. $\frac{1}{10}$

What is the distribution for the length of time one battery lasts?

What is the distribution for the mean length of time 64 batteries last?

N

What is the distribution for the total length of time 64 batteries last?

Find the probability that the sample mean is between seven and 11.

0.7799

Find the 80 th percentile for the total length of time 64 batteries last.

Find the IQR for the mean amount of time 64 batteries last.

1.69

Find the middle 80% for the total amount of time 64 batteries last.

Use the following information to answer the next eight exercises:
A uniform distribution has a minimum of six and a maximum of ten. A sample of 50 is taken.

Find P ( Σx >420).

0.0072

Find the 90 th percentile for the sums.

Find the 15 th percentile for the sums.

391.54

Find the first quartile for the sums.

Find the third quartile for the sums.

405.51

Find the 80 th percentile for the sums.

#### Questions & Answers

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Source:  OpenStax, Statistics i - math1020 - red river college - version 2015 revision a - draft 2015-10-24. OpenStax CNX. Oct 24, 2015 Download for free at http://legacy.cnx.org/content/col11891/1.8
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