Download lect5-4 - Rutgers Statistics

Normal Distribution as an Approximation
to the Binomial Distribution
Section 5-6
M A R I O F. T R I O L A
Copyright © 1998, Triola, Elementary Statistics
Copyright © 1998, Triola, Elementary Statistics
Addison
Wesley
Longman
Addison
Wesley Longman
1
Review
Binomial Probability Distribution
applies 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 Longman
2
Approximate a Binomial Distribution
with a Normal Distribution if:
1. np  5
2. nq  5
Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman
3
Approximate a Binomial Distribution
with a Normal Distribution if:
1. np  5
2. nq  5
Then µ = np and s =
npq
and the random variable has
a
distribution.
(normal)
Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman
4
Figure 5-24 Solving Binomial Probability
Problems Using a Normal Approximation
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Addison Wesley Longman
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Figure 5-24 Solving Binomial Probability
Problems Using a Normal Approximation
Start
1
2
3
After verifying that we have a binomial
probability problem, identify n, p, q
4
Is
Computer Software
Available ?
Yes
Use the
Computer Software
No
Can the
problem be solved
by using Table A-1
?
Yes
Use the Table A-1
No
Can the
problem be easily solved
with the binomial
probability formula
?
Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman
Yes
Use binomial probability formula
n!
P(x) = (n – x)!x!• p x • q
n–x
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Figure 5-24 Solving Binomial Probability
Problems Using a Normal Approximation
Can the
problem be easily solved
with the binomial
probability formula
?
4
5
6
7
Yes
Use binomial probability formula
n!
P(x) = (n – x)!x!• p x • q n–x
No
Are np  5 and
nq  5
both true ?
No
Yes
Compute µ = np and s = npq
Draw the normal curve, and identify the region
representing the probability to be found. Be sure
to include the continuity correction. (Remember,
the discrete value x is adjusted for continuity by
adding and subtracting 0.5)
Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman
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Figure 5-24 Solving Binomial Probability
Problems Using a Normal Approximation
Draw the normal curve, and identify the region
representing the probability to be found. Be sure
to include the continuity correction. (Remember,
the discrete value x is adjusted for continuity by
adding and subtracting 0.5)
Calculate
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8
9
z =x – µ
s
where µ and s are the values already found and
x is adjusted for continuity.
Refer to Table A-2 to find the area between µ and
the value of x adjusted for continuity. Use that area
to determine the probability being sought.
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Addison Wesley Longman
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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
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Continuity Corrections Procedures
continued
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
itself is to be included, The total shaded region corresponds to
probability being sought.
Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman
x
10
x = at least 64
.
= 64, 65, 66, . . .
50
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64
63.5
11
x = at least 64
.
= 64, 65, 66, . . .
50
64
63.5
x = more than 64
= 65, 66, 67, . . .
50
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Addison Wesley Longman
65
64.5
12
x = at least 64
.
= 64, 65, 66, . . .
50
64
63.5
x = more than 64
= 65, 66, 67, . . .
50
x = at most 64
= 0, 1, . . . 62, 63, 64
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Addison Wesley Longman
65
64.5
50 64
64.5
13
x = at least 64
.
= 64, 65, 66, . . .
50
64
63.5
x = more than 64
= 65, 66, 67, . . .
50
x = at most 64
= 0, 1, . . . 62, 63, 64
65
64.5
50 64
64.5
x = fewer than 64
= 0, 1, . . . 62, 63
50 63
63.5
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x = exactly
64
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15
x = exactly
64
50
64
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x = exactly
64
50
64
50
63.5 64.5
Interval represents discrete number 64
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Addison Wesley Longman
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Chapter 5
Normal Probability Distributions
5-1 Overview
5-2 The Standard Normal Distribution
5-3 & 5-4 Nonstandard Normal Distributions
(Finding Probabilities & Finding Scores)
5-5 The Central Limit Theorem
5-6 Normal Distributions as Approximation
to Binomial Distribution
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Addison Wesley Longman
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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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Addison Wesley Longman
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