chapter 7 npd

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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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