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Chapter SevenThe Normal Probability The Normal Probability DistributionDistributionGOALS
When you have completed this chapter, you will be able to:ONEList the characteristics of the normal probability distribution.
TWO Define and calculate z values.
THREEDetermine the probability an observation will lie between two points using the standard normal distribution.
FOURDetermine the probability an observation will be above or below a given value using the standard normal distribution.
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Time of arrivals between two consecutive customers in a retail store.
Height of students in a class.
Pressure of car tire.
Time taken to reach IIUM campus from your house
Examples of continuous random variables
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Various Continuous DistributionsVarious Continuous Distributions
Uniform distribution Normal distribution Exponential distribution Gamma Distribution Beta distribution Chi-square distribution t –distribution F distribution
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Normal DistributionNormal Distribution
Normal distribution is perhaps the most important from amongst all the continuous distributions. This is because a large number of physical phenomena follow normal distribution.
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Suppose, in a college, there are 3,264 male students. Their mean height and S.D. of heights are respectively, 64.4 and 2.4 inches.
The following table provided the frequency distribution of all the male students:
An Example:
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Height (inches) Frequency Relative frequency
56-57 3 0.0009
57-58 6 0.0018
58-59 26 0.0080
59-60 74 0.0227
60-61 147 0.0450
61-62 247 0.0757
62-63 382 0.1170
63-64 483 0.1480
64-65 559 0.1713
Frequency distribution of heights of college students
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Height (inches) Frequency Relative frequency
65-66 514 0.1575
66-67 359 0.1170
67-68 240 0.0735
68-69 122 0.0374
69-70 65 0.0199
70-71 24 0.0074
71-72 7 0.0021
72-73 5 0.0015
73-74 1 0.0003
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0 . 4
0 . 3
0 . 2
0 . 1
. 0
x
f(
x
r a l i t r b u i o n : = 0 , = 1
Characteristics of a Normal Distribution
Mean, median, andmode are equal
Normalcurve issymmetrical
Theoretically,curveextends toinfinity
a
McGraw-Hill/Irwin Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved
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Some more examples of normal Some more examples of normal random variable:random variable:
Students aptitude test scores in some test ,e.g. GRE, GMAT, TOFEL, etc.
Weight of people. Years of life expectancy. Most of the items produced or filled by machines.
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Approximate Normal DistributionApproximate Normal Distribution
In reality, most of the normal variables are actually “approximately normal”.
Though they are approximately normal, but still we apply the theory of normal distribution.
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718.2
14.3
.
2
1)(
2
2
2
e
DS
meanwhere
exf
x
Family of Normal Distributions
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Normal curves having same Normal curves having same mean but different S.D.smean but different S.D.s
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With continuous distribution, probabilities of outcomes occurring between particular points are determined by calculating the area under the curve between those points.
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For plant B, what is the probability that a randomly selected employee’s length of service will be between 22 and 25 years?
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dxe
dxxf
x2
2
2
25
22
25
22
2
1
)(
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There are infinite number of normal distributions for infinite number of combination of values of (µ and σ).
Fortunately, we can transform all normal distributions to a single normal distribution,
called standard normal distribution.
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Standard Normal VariableStandard Normal Variable
X
z
1. Above is the way to standardize all the normal variables.
2. Z represents number of S.D.s away from the mean.
3. Standard normal variable has zero mean and S.D. = 1.
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Previous ExamplePrevious Example
28.19.3
2025,25
51.09.3
2022,22
zXFor
XzXFor
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Example 1: Let us state the previous problem again. The mean length of
service of the employees in Plant B = 20 years with S.D. = 3.9 years. What is the probability that a randomly selected employee’s length of service lies between mean and 22 years?
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Example 2: What is the probability that a randomly selected employee’s length of service is less than 22 years? [z=0.51, p=0.195+0.50 = 0.695]
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Example 3: What is the probability of obtaining a length of service
greater than 26 years? [Ans: z=1.54; p=0.5-0.4382=0.0618]
Example 4: What is the probability that the length of service will be less
than 19 years? [Ans: z=-0.26; p=0.50-0.1026=0.3974]
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Example 5: What is the probability that the length of service lie between
16 and 23 years? [Ans: z=-1.03, 0.77; 0.3485+0.2794=0.6279]
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Example 6: What is the probability that the length of service will lie between 17-19 years? [z=-0.77, z=0.26; p=0.2794-0.1026=0.1768]
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Application of Empirical RuleApplication of Empirical Rule
-3 -2 -1 +1 +2 +3Mean
68.26%95.44%99.74%
= Standard deviation
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Example 7: The Federal Reserve System publishes data on family income
based on its Survey of Customer Finances. When the head of the household has a college degree, the mean before-tax family income is $70,400. Suppose that 60% of the before-tax family incomes when the head of the household has a college degree are between $61,200 and $79,600 and that these incomes are normally distributed. What is the standard deviation of before-tax family incomes when the head of the household has a college degree?
[area=0.84 for z=0.3; sigma=10952]
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Example 8: A tire manufacturer wishes to set a minimum mileage guarantee
on its new MX 100 tire. Tests reveal the mean mileage is 68,500 with a standard deviation of 2125 miles and a normal distribution. The manufacturer wants to set the minimum guaranteed mileage so that no more than 5 percent of the tires will have to be replaced. What minimum guaranteed mileage should the manufacturer announce?
[z=-1.645, X=65004]
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