copyright © 1998, triola, elementary statistics addison wesley longman 1 normal distribution as an...

Copyright © 1998, Triola, Elementary Statistics

Addison Wesley Longman1

Normal Distribution as an Normal Distribution as an ApproximationApproximation

to the Binomial Distributionto the Binomial DistributionSection 5-6Section 5-6

M A R I O F. T R I O L ACopyright © 1998, Triola, Elementary Statistics

Addison Wesley Longman



Review Binomial Probability Distributionapplies to a discrete random variable

has these requirements:

1. The experiment must have fixed number of trials.

2. The trials must be independent.

3. Each trial must have all outcomes classified into two categories.

4. The probabilities must remain constant for each trial.

solve by P(x) formula, computer software, or Table A-1



Approximate a Binomial Distributionwith a Normal Distribution if:

1. np 5

2. nq 5



Approximate a Binomial Distributionwith a Normal Distribution if:

1. np 5

2. nq 5

distribution.(normal)

Then µ = np and = npq

and the random variable has

a



Figure 5-24 Solving Binomial Probability Problems Using a Normal Approximation




After verifying that we have a binomialprobability problem, identify n, p, q

Is Computer Software

Available ?

Can the problem be solved by using Table A-1

?

Can the problem be easily solved

with the binomial probability formula

?

Use theComputer Software

Use the Table A-1

Use binomial probability formulaYes

Yes

Yes

No

No

Start

P(x) = • p

x • q(n – x)!x!n!

12

3

4

n–x




Can the problem be easily solved

with the binomial probability formula

?

Use binomial probability formulaYes

P(x) = • p

x • q(n – x)!x!n!

7654

Are np 5 andnq 5

both true ?

No

No

Yes

Compute µ = np and = npq

Draw the normal curve, and identify the regionrepresenting the probability to be found. Be sureto include the continuity correction. (Remember, the discrete value x is adjusted for continuity byadding and subtracting 0.5)

n–x




89

7

Draw the normal curve, and identify the regionrepresenting the probability to be found. Be sureto include the continuity correction. (Remember, the discrete value x is adjusted for continuity byadding and subtracting 0.5)

Calculate

where µ and are the values already found and x is adjusted for continuity.

z = x – µ

Refer to Table A-2 to find the area between µ and the value of x adjusted for continuity. Use that areato determine the probability being sought.



Continuity Corrections Procedures

1. When using the normal distribution as an approximation to the binomial distribution, always use the continuity correction.

2. In using the continuity correction, first identify the discrete whole number

x that is relevant to the binomial probability problem.

3. Draw a normal distribution centered about µ, then draw a vertical strip

area centered over x . Mark the left side of the strip with the number x

0.5, and mark the right side with x + 0.5. For x = 64, draw a strip from 63.5 to 64.5. Consider the area of the strip to represent the probability

of discrete number x.

continued



Continuity Corrections Procedures

4. Now determine whether the value of x itself should be included in the probability you want. Next, determine whether you want the

probability of at least x, at most x, more than x, fewer than x, or

exactly x. Shade the area to the right of left of the strip, as

appropriate; also shade the interior of the strip itself if and only if x itself is to be included, The total shaded region corresponds to probability being sought.

continued



x = at least 64 = 64, 65, 66, . . .

645063.5

.



x = at least 64 = 64, 65, 66, . . .

645063.5

x = more than 64 = 65, 66, 67, . . .

655064.5

.



x = at least 64 = 64, 65, 66, . . .

645063.5

x = more than 64 = 65, 66, 67, . . .

x = at most 64 = 0, 1, . . . 62, 63, 64

645064.5

655064.5

.



x = at least 64 = 64, 65, 66, . . .

645063.5

x = more than 64 = 65, 66, 67, . . .

x = at most 64 = 0, 1, . . . 62, 63, 64

x = fewer than 64 = 0, 1, . . . 62, 63

645064.5

635063.5

655064.5

.



x = exactly 64



6450

x = exactly 64



Interval represents discrete number 64

6450

64.563.5

50

x = exactly 64



Chapter 5Normal Probability Distributions

5-1 Overview5-2 The Standard Normal Distribution5-3 & 5-4 Nonstandard Normal Distributions

(Finding Probabilities & Finding Scores)

5-5 The Central Limit Theorem5-6 Normal Distributions as

Approximation to Binomial Distribution



Basic Concepts• Continuous distribution/Density curve

• Uniform distribution

• Normal distribution– Standard normal distribution

• Central Limit Theorem (Approx. normal distr.)– Distribution of sample mean

• mean, variance, standard deviation (standard error)

– finite population correction factor– continuity correction (Binomial distribution)

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