Transcript z - McGraw Hill Higher Education

Statistics
A First Course
Donald H. Sanders
Robert K. Smidt
Aminmohamed Adatia
Glenn A. Larson
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Chapter 5
Probability Distributions
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Chapter 5 - Topics
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Binomial Experiments
Determining Binomial Probabilities
The Poisson Distribution
The Normal Distribution
Normal Approximation of the Binomial
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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
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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
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Our QuickQuiz probability distribution.
Figure 5.1 (including table)
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Expected Value (Mean) of Binomial Distribution Formula
Variance of Binomial Distribution Formula
Standard Deviation of Binomial Distribution Formula
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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
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The Normal Distribution
• Continuous probability distribution
• Used to investigate the probability that the
variable assumes any value within a given
interval of values
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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 the
mean (z = 1.00).
Figure 5.6
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Calculating Probabilities for the
Standard Normal Distribution
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
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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
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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
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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
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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
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The z score interval corresponding to
70 < x < 130
Figure 5.13
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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
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90th Percentile of z scores
Figure 5.20
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The Normal Approximation
of the Binomial
Graph showing both the binomial
probability histogram and the
normal distribution
Figure 5.13
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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
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End of Chapter 5
Probability Distributions
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