Transcript LO7-3 - McGraw Hill Higher Education - McGraw

Continuous
Probability
Distributions
Chapter 7
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Learning Objectives
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LO7-2 Describe the characteristics of a normal probability distribution.
LO7-3 Describe the standard normal probability distribution and use
it to calculate probabilities.
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LO7-2 Describe the characteristics of a
normal probability distribution.
Characteristics of a Normal
Probability Distribution
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It is bell-shaped and has a single peak at the center of the distribution.
It is symmetrical about the mean.
It is asymptotic: The curve gets closer and closer to the X-axis but never
actually touches it. To put it another way, the tails of the curve extend
indefinitely in both directions.
The location of a normal distribution is determined by the mean, . The
dispersion or spread of the distribution is determined by the standard
deviation, σ.
The arithmetic mean, median, and mode are equal.
As a probability distribution, the total area under the curve is defined to be
1.00.
Because the distribution is symmetrical about the mean, half the area under
the normal curve is to the right of the mean, and the other half to the left of it.
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LO7-2
The Normal Distribution –
Graphically
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LO7-2
The Family of Normal
Distributions
Equal Means and Different
Standard Deviations
Different Means and
Standard Deviations
Different Means and Equal Standard Deviations
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LO7-3 Describe the standard normal probability
distribution and use it to calculate probabilities.
The Standard Normal Probability
Distribution
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The standard normal distribution is a normal
distribution with a mean of 0 and a standard
deviation of 1.
It is also called the z distribution.
A z-value is the signed distance between a
selected value, designated x, and the population
mean, , divided by the population standard
deviation, σ.
The formula is:
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LO7-3
Areas Under the Normal Curve
Using a Standard Normal Table
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LO7-3
The Empirical Rule – Verification
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For z=1.00, the table’s
value is 0.3413; times 2
is 0.6826.
For z=2.00, the table’s
value is 0.4772; times 2
is 0.9544.
For z=3.00, the table’s
value is 0.4987; times 2
is 0.9974.
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LO7-3
The Normal Distribution –
Example
The weekly incomes of shift
foremen in the glass
industry follow the normal
probability distribution with
a mean of $1,000 and a
standard deviation of
$100.
What is the z value for the
income, let’s call it x, of a
foreman who earns $1,100
per week? For a foreman
who earns $900 per week?
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LO7-3
Normal Distribution – Finding
Probabilities (Example 1)
In an earlier example we
reported that the mean
weekly income of a shift
foreman in the glass
industry
is
normally
distributed with a mean of
$1,000 and a standard
deviation of $100.
What is the likelihood of
selecting
a
foreman
whose weekly income is
between $1,000 and
$1,100?
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LO7-3
Normal Distribution – Finding
Probabilities (Example 1)
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LO7-3
Finding Areas for z Using Excel
The Excel function:
=NORM.DIST(x,Mean,Standard_dev,Cumu)
=NORM.DIST(1100,1000,100,true)
calculates the probability (area) for z=1.
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LO7-3
Normal Distribution – Finding
Probabilities (Example 2)
Refer to the information
regarding
the
weekly
income of shift foremen in
the glass industry. The
distribution
of
weekly
incomes follows the normal
probability distribution with a
mean of $1,000 and a
standard deviation of $100.
What is the probability of
selecting a shift foreman in
the glass industry whose
income is between $790
and $1,000?
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LO7-3
Normal Distribution – Finding
Probabilities (Example 3)
Refer to the information
regarding
the
weekly
income of shift foremen in
the glass industry. The
distribution
of
weekly
incomes follows the normal
probability distribution with a
mean of $1,000 and a
standard deviation of $100.
What is the probability of
selecting a shift foreman in
the glass industry whose
income is less than $790?
The probability of selecting a shift
foreman with income less than
$790 is 0.5 - .4821 = .0179.
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LO7-3
Normal Distribution – Finding
Probabilities (Example 4)
Refer to the information
regarding
the
weekly
income of shift foremen in
the glass industry. The
distribution
of
weekly
incomes follows the normal
probability distribution with
a mean of $1,000 and a
standard deviation of $100.
What is the probability of
selecting a shift foreman in
the glass industry whose
income is between $840
and $1,200?
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LO7-3
Normal Distribution – Finding
Probabilities (Example 5)
Refer to the information
regarding
the
weekly
income of shift foremen in
the glass industry. The
distribution
of
weekly
incomes follows the normal
probability distribution with
a mean of $1,000 and a
standard deviation of $100.
What is the probability of
selecting a shift foreman in
the glass industry whose
income is between $1,150
and $1,250?
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