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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
Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman2
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
Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman3
Approximate a Binomial Distributionwith a Normal Distribution if:
1. np 5
2. nq 5
Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman4
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
Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman5
Figure 5-24 Solving Binomial Probability Problems Using a Normal Approximation
Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman6
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
Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman7
Figure 5-24 Solving Binomial Probability Problems Using a Normal Approximation
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
Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman8
Figure 5-24 Solving Binomial Probability Problems Using a Normal Approximation
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.
Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman9
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
Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman10
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
Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman11
x = at least 64 = 64, 65, 66, . . .
645063.5
.
Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman12
x = at least 64 = 64, 65, 66, . . .
645063.5
x = more than 64 = 65, 66, 67, . . .
655064.5
.
Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman13
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
.
Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman14
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
.
Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman15
x = exactly 64
Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman16
6450
x = exactly 64
Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman17
Interval represents discrete number 64
6450
64.563.5
50
x = exactly 64
Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman18
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
Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman19
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)