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© 2005 McGraw-Hill Ryerson Ltd. 5-1
StatisticsA First Course
Donald H. SandersRobert K. Smidt
Aminmohamed AdatiaGlenn A. Larson
© 2005 McGraw-Hill Ryerson Ltd. 5-2
Chapter 5
Probability Distributions
© 2005 McGraw-Hill Ryerson Ltd. 5-3
Chapter 5 - Topics
• Binomial Experiments• Determining Binomial Probabilities• The Poisson Distribution• The Normal Distribution• Normal Approximation of the Binomial
© 2005 McGraw-Hill Ryerson Ltd. 5-4
Binomial Experiments
• Properties of a Binomial Experiment– Same action (trial) is repeated a fixed
number of times– Each trial is independent of the others– Two possible outcomes – success or failure– Constant probability of success for each trial
© 2005 McGraw-Hill Ryerson Ltd. 5-5
Determining Binomial Probabilities
• Combinations– Selection of r items from a set of n distinct
objects without regard to the order in which r items are picked
Combination Rule
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Determining Binomial Probabilities
• Binomial Probability– Probability of correctly guessing exactly r items
from a set of n distinct objects without regard to the order in which r items are picked
Binomial Probability Formula
© 2005 McGraw-Hill Ryerson Ltd. 5-7
Our QuickQuiz probability distribution.
Figure 5.1 (including table)
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© 2005 McGraw-Hill Ryerson Ltd. 5-9
Variance of Binomial Distribution Formula
Standard Deviation of Binomial Distribution Formula
Expected Value (Mean) of Binomial Distribution Formula
© 2005 McGraw-Hill Ryerson Ltd. 5-10
The Poisson Distribution
• Discrete probability distribution• Used to determine the number of specified
occurrences that take place within a unit of time, distance, area, or volume
Poisson Distribution Formula
© 2005 McGraw-Hill Ryerson Ltd. 5-11
The Normal Distribution
• Continuous probability distribution• Used to investigate the probability that the
variable assumes any value within a given interval of values
© 2005 McGraw-Hill Ryerson Ltd. 5-12
Normal Distribution.
Figure 5.4
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Probability of breaking strength between 110 and 120.
Figure 5.5
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Both intervals extend from the mean (z = 0) to 1 standard
deviation above themean (z = 1.00).
Figure 5.6
© 2005 McGraw-Hill Ryerson Ltd. 5-18
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The probability that a z value selected at random will fall between
0 and 2.27 or between–2.27 and 0 is .4884.
Figure 5.7
Calculating Probabilities for the Standard Normal Distribution
© 2005 McGraw-Hill Ryerson Ltd. 5-20
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The area under the normal curve between vertical lines drawn at
z = –1.73 and z = +2.45 is .9511.
Figure 5.8
© 2005 McGraw-Hill Ryerson Ltd. 5-22
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The area under the normal curve between a z value of –1.54 and
a z value of –.76 is .1618.
Figure 5.9
© 2005 McGraw-Hill Ryerson Ltd. 5-24
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The area under the normal curve to the left of a z value of
–1.96 is .0250.
Figure 5.10
© 2005 McGraw-Hill Ryerson Ltd. 5-26
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The area under the normal curve to the left of a z value
of 1.42 is .9222.
Figure 5.11
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The Normal Distribution• Computing Probabilities for Any Normally Distributed
Variable– z scores correspond to the number of standard deviations a
data value is from the mean– Any value can be converted to a standard score (z score)
Convert x value to z score formula
© 2005 McGraw-Hill Ryerson Ltd. 5-30
The z score interval corresponding to 70 < x < 130
Figure 5.13
© 2005 McGraw-Hill Ryerson Ltd. 5-31
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The Normal Distribution
• Finding Cut-off Scores for Normally Distributed Variables – Given the area under the standard normal curve, the z
score method can be used to calculate the cut off point
Convert z score to x value formula
© 2005 McGraw-Hill Ryerson Ltd. 5-33
90th Percentile of z scoresFigure 5.20
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Graph showing both the binomial probability histogram and the
normal distribution
Figure 5.13
The Normal Approximation of the Binomial
© 2005 McGraw-Hill Ryerson Ltd. 5-36
© 2005 McGraw-Hill Ryerson Ltd. 5-37
The Normal Approximation of the Binomial
• Computing Probabilities for Any Normally Distributed Variable Method– Calculate mean and standard deviation– Apply continuity correction factor (±0.5)– Convert x values to z scores– Calculate area under standard normal curve
© 2005 McGraw-Hill Ryerson Ltd. 5-38
End of Chapter 5
Probability Distributions
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