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Central Limit Theorem: Approximating Distributions with Large Sample Sizes - Prof. John Bu, Study notes of Statistics

The central limit theorem, which states that as the sample size increases, the distribution of sample means approaches a normal distribution, regardless of the distribution of the original data. It covers practical rules for applying the theorem, examples, and the use of normal distribution to approximate binomial distribution. The document also includes an example of finding probabilities with a normally distributed population.

Typology: Study notes

Pre 2010

Uploaded on 07/30/2009

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Sierra College – Math 13
Spring 2009 – Class 16/32
Today: Sections 6-5/6-6; Homework #2 Due
Assignment: 6-5 {1, 3, 5, 7, 11, 15, 19}
6-6 {1, 3, 5, 7, 15, 17, 19, 23, 25}
Next: Test #2
Instructor: John Burke
E-mail: john_burke@mindspring.com
Web Page: http://math.sierracollege.edu/Staff/JohnBur ke/
Telephone: 916 337-0425
Office hours: (V-307) MW 2:35-5:00; M 2:45-3:45 (official)
2
6-5 The Central Limit Theorem
The random variable xhas a distribution (which may or may
not be normal) with mean µ and standard deviation σ; and,
Samples all of the same size n are randomly selected from
the population of all x values.
Then,
Given that
3
6-5 The Central Limit Theorem
The distribution of sample means will, as the sample size
increases, approach a normal distribution.
The mean of the sample means ( ) will be the population
mean µ.
The standard deviation of the sample means ( ) will
approach σ .
n
x
µ
x
σ
pf3
pf4
pf5

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1

Sierra College – Math 13

Spring 2009 – Class 16/

Today: Sections 6-5/6-6; Homework #2 Due

Assignment: 6-5 {1, 3, 5, 7, 11, 15, 19} 6-6 {1, 3, 5, 7, 15, 17, 19, 23, 25} Next: Test #

Instructor: John Burke

E-mail: john_burke@mindspring.com Web Page: http://math.sierracollege.edu/Staff/JohnBurke/

Telephone: 916 337-

Office hours: (V-307) MW 2:35-5:00; M 2:45-3:45 (official)

2

6-5 The Central Limit Theorem

The random variable x has a distribution (which may or may not be normal) with mean μ and standard deviation σ; and,

Samples all of the same size n are randomly selected from the population of all x values.

Then,

Given that

6-5 The Central Limit Theorem

The distribution of sample means will, as the sample size increases, approach a normal distribution.

The mean of the sample means ( ) will be the population mean μ.

The standard deviation of the sample means ( ) will approach σ.

n

μ x

σ x

4

6-5 The Central Limit Theorem

Practical Rules Commonly Used

  1. For samples of size n > 30, the distribution of the sample means can be approximated reasonably well by a normal distribution. The approximation gets better as the sample size n becomes larger.
  2. If the original population is itself normally distributed, then the sample means will be normally distributed for any sample size n (not just for values of n larger than 30).

5

−0.5 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.

6-5 The Central Limit Theorem

(^1) ( 1 ) 2 2

x

e

y

− − .

(^1) ( 1 ) 2 2

x

e

y

π

− − .

Suppose μ=1.0; σ=0.

Then the distribution of sample means has μ=1.0; σ=0.1; n=

6

μ = 143 150 σ = 29

(^0) z

Example : Given that the population of women has normally distributed weights with a mean of 143 lb and a standard deviation of 29 lb, a.) if one woman is randomly selected, find the probability that her weight is greater than 150 lb.

P(x > 150) =?

10

When working with an individual value from a normally distributed population use the methods of section 5.3; i.e.,

When working with a mean for some sample or group, be sure to use the value of for the standard deviation of the sample means and use the formula

n

σ

x

z

n

μ σ

6-5 The Central Limit Theorem

Review

x

z

11

Maximum Capacity = 18 people or 3000 pounds. The capacity will be exceeded if 18 people have weights with a mean greater than 3000/18 = 167 pounds. Men have weights that are normally distributed with a mean of 172 and a standard deviation of 29 lb.

  1. Find the probability that if an individual man is randomly selected, his weight will be greater than 167 pounds.
  2. Find the probability that 18 randomly selected men will have a mean weight that is greater than 167 pounds (so that their total weight is greater than the elevator capacity of 3000 lb.

Example: V-Building Elevator

If np ≥ 5 and nq ≥ 5 , then μ = np and σ = npq and

the random variable has a

6-6 Approximating a

Binomial Distribution

with a Normal Distribution

(normal) distribution.

13

6-6 Procedure for Using a Normal

Distribution to Approximate a

Binomial Distribution

  1. Try to solve directly via table, technology or formula. If impractical, then
  2. Establish that the normal distribution is a suitable approximation to the binomial distribution by verifying np ≥ 5 and nq ≥ 5.
  3. Find the values of the parameters μ and σ by calculating μ = np and σ = npq.
  4. Identify the discrete value of x (the number of successes). Change the discrete value x by replacing it with the interval from x - 0.5 to x + 0. (continuity correction). Draw a normal curve and enter the values of μ and σ.
  5. Change x by replacing it with x - 0.5 or x + 0.5, as appropriate.
  6. Find the area corresponding to the desired probability.

14

Example: Gender Discrimination

The president of Portland College finds that applicants for admission are evenly divided between men and women, and both genders are equally qualified. She concludes that accepted students should be roughly 50% men and 50% women. She checks and finds that among the 1000 students accepted last year, 520 are men and only 480 women.

Find the probability of selecting at least 520 men, assuming that the large pool of applicants consists of equal numbers of men and women. Does there appear to be discrimination based upon gender?

P(at least 520 men) = P(x=520) + P(x=521) +…+ P(x=1000)

Impractical if we have to do it manually!

Finding the Probability of “At Least” 520

Men Among 1000 Accepted Applicants

Step 1: Verify that it is reasonable to approximate the binomial distribution. np = 1000 x 0.5 = 500 & nq = 1000 x 0.5 = 500 So, np and nq are both ≥ 5.

0

520

z