This document describes the characteristics of a normal probability distribution, including that it is bell-shaped and symmetrical about the mean. It then discusses the standard normal distribution, which has a mean of 0 and standard deviation of 1, and how it can be used to calculate probabilities. Several examples are provided to demonstrate how to find probabilities for different income ranges using the standard normal distribution, given data about the mean and standard deviation of shift foremen incomes.

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Continuous
Probability
Distributions
Chapter 7

2. 7-2
Learning Objectives
 LO7-2 Describe the characteristics of a normal probability distribution.
 LO7-3 Describe the standard normal probability distribution and use
it to calculate probabilities.

3. 7-3
Characteristics of a Normal
Probability Distribution
 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.
LO7-2 Describe the characteristics of a
normal probability distribution.

4. 7-4
The Normal Distribution –
Graphically
LO7-2

5. 7-5
The Family of Normal
Distributions
Different Means and
Standard Deviations
Equal Means and Different
Standard Deviations
Different Means and Equal Standard Deviations
LO7-2

6. 7-6
The Standard Normal Probability
Distribution
 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:
LO7-3 Describe the standard normal probability
distribution and use it to calculate probabilities.

7. 7-7
Areas Under the Normal Curve
Using a Standard Normal Table
LO7-3

8. 7-8
The Empirical Rule – Verification
 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.
LO7-3

9. 7-9
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?
9
LO7-3

10. 7-10
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?
LO7-3

11. 7-11
Normal Distribution – Finding
Probabilities (Example 1)
LO7-3

12. 7-12
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.
LO7-3

13. 7-13
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?
LO7-3

14. 7-14
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?
LO7-3
The probability of selecting a shift
foreman with income less than
$790 is 0.5 - .4821 = .0179.

15. 7-15
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?
LO7-3

16. 7-16
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?
LO7-3