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Fox/Levin/Forde, Elementary Statistics in Social Research, 12e
Chapter 5: Probability and the Normal
Curve
HLTH 300 Biostatistics for Public Health
Practice,
Raul Cruz-Cano, Ph.D
2/24/2014, Spring 2014
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© 2014 by Pearson Higher Education, Inc
Upper Saddle River, New Jersey 07458 • All Rights Reserved
Announcements
• Okay, let’s have a review before the exams…but we
need to reduce the number of exams
(4 Exams+ 4 Reviews = 8 Sessions = .5 of our meetings!)
• See new syllabus
• Rule for the reviews:
– Specific questions, not “blanket” questions.
– Not obligatory to attend.
• For the exams you are allowed to bring copies of
Appendix C and Appendix D
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CHAPTER OBJECTIVES
5.1
Calculate probabilities and understand the rules of probability
5.2
Understand the concept of a probability distribution
5.3
List the characteristics of the normal curve
5.4
Understand the area under the normal curve
5.5
Calculate and use z scores
© 2014 by Pearson Higher Education, Inc
Upper Saddle River, New Jersey 07458 • All Rights Reserved
Learning Objectives
After this lecture, you should be able to complete the following Learning Outcomes
5.1
Calculate probabilities and
understand the rules of probability
5.1
Probability
The relative likelihood of occurrence of any given
outcome
P=0
P = .5
P=1
The outcome is
Impossible
The outcome is
as likely to
happen as not
happen
The outcome is
certain
5.1
The Rules of Probability
P (F) 
Number of times the outcome or event can occur
Total number of times any outcome or event can occur
Converse Rule: The probability that something will not occur
 
P F  1  P F 
Addition Rule: The probability of obtaining one of several different and
distinct outcomes (mutually exclusive)
P  A or B  P  A   P B
Multiplication Rule: The probability of obtaining two or more outcomes in
combination (Independent)
P  A and B  P  A  X P B
5.1
The Rules of Probability
Addition Rule: The probability of obtaining one of several different
outcomes (not mutually exclusive)
P  A or B  P  A   P B -P (A and B)
Example
A= Intoxicated with C02
B= Intoxicated with N02
1.
2.
3.
4.
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What is P(A)? P(B)?
What is P(Ā)?
What is P(A and B)?
What is P(A or B)
5.1
Probability: Example
Heads or Tails?
1
 .5
2
1
 .5
2
.5 .5   .25
5.1
Probability: Example
Heads or Tails?
P HT   P  TH  P H P  T   P  T  P H
 .50 .50   .50 .50 
 .25  .25
 .50
Example
In the U.S. the probability of a driver being uninsured is .123
1. Two drivers crash:
1. P(Both uninsured)
2. P(Both Insured)
2. Five Drivers Crash, P(all uninsured)?
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Learning Objectives
After this lecture, you should be able to complete the following Learning Outcomes
5.2
Understand the concept of a
probability distribution
5.2
Probability Distributions
Directly analogous to a frequency distribution
• Except it is based on probability theory
Mean = μ
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Standard
Deviation = σ
5.2
X = # of heads
P
0
.25
1
.50
2
.25
Total
1.00
Figure 5.1
Examples
Problem 6:
Standard 6-sided die:
X: Outcome of roll
a. P(X=2)
b. P(X=3 or X=4)
c. P(X=Odd Number)
d. P(X=Anything but 5)
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Examples
Problem 14
X: A random politician from the sample
Pol.
Party
Support
1
R
No
2
D
No
a.
P(X=Republican)
b.
P(X=Democrat that support
euthanasia)
3
D
Yes
c.
P(X=Does not support euthanasia)
4
R
No
d.
P(X=Republican that does not
support euthanasia)
5
R
No
6
D
No
7
D
Yes
8
D
Yes
9
R
No
10
D
Yes
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Examples
Problem 20
Lottery ticket 2 numbers and a letter, e.g. 3 7 P
a. P(match 1st digit)
b. P(match 2nd digit)
c. P(Not match 1st digit)
d. P(match 1st and 2nd digit)
e. P(match letter)
f. P(Perfect match)
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Learning Objectives
After this lecture, you should be able to complete the following Learning Outcomes
5.3
List the characteristics of the normal
curve
5.3
•
•
•
•
•
•
•
•
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Characteristics of the Normal Curve
Smooth
Symmetrical
Unimodal
Mean = Median = Mode
Infinite in Both Directions
Probability Distribution
Mean = μ; Standard Deviation = σ
Areas Under the Curve = 100%
Characteristics of the Normal Curve
If you know the mean and the std. deviation of a normally distributed
variable then you can find many probabilities:
P(X < x )
P(X > x )
P(x1 <X < x2)
If you know the mean and the std. deviation of the grades of exams for a
class (normally distributed) then you can find many probabilities:
X: Grade of a random student
P(X < 90 )
P(X > 85 )
P(75 < X < 95
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5.3
The Reality of the Normal Curve
The normal curve is a theoretical ideal
Many many variables do conform to the normal curve…
few students get low grades, few students get great
grades, most get around average
Some variables do not conform to the normal curve
• Many distributions are skewed, multi-modal, and symmetrical but
not bell-shaped
• Assuming normality when it does not exist can impact the validity
of our conclusions
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Learning Objectives
After this lecture, you should be able to complete the following Learning Outcomes
5.4
Understand the area under the
normal curve
5.4
Figure 5.5
Like a very smooth histogram
5.4
Figure 5.6
5.4
Figure 5.7
5.4
Figure 5.8
Learning Objectives
After this lecture, you should be able to complete the following Learning Outcomes
5.5
Calculate and use z scores
5.5
Standard Scores and the Normal Curve
It is possible to determine the area under the curve for
any sigma distance from the mean
This distance is called a z-score
• Indicates direction and distance that any raw score deviates
from the mean in sigma units
z
X 

  mean of a distribution
  standard deviation of a distribution
z  standard score
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Example
X: Salaries in a company
Mean= $20,000
Std. Dev=$1,500
Distribution = Normal
P(20,000 < X < 22,000)=?
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P(20,000 < X < 22,000) =40.82%
Exercises:
1. P(X < 22,000)
2. P(X < 22,000)
3. P(18,000 < X < 22,000)
4. P(X<18,000)
5. P(X>18,000)
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5.5
Finding Probability under the Normal
Curve
When the normal curve is used in conjunction with z
scores and Table A in Appendix C, we can determine
the probability of obtaining any raw score (X) in a
distribution
• The converse, addition, and multiplication rules still apply
We can also reverse this process to calculate score
values from particular portions of area or percentages
X    z
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X:911 Response time
Mean= 5.6 minutes
Std. Dev= 1.8 minutes
Distribution = Normal
P(X<x)=75%
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Examples
Problem 27
X: SAT Scores
Normally distributed
Mean=500
Std. Dev. = 100
a. P(500< X <600)
b. P(400 < X < 600)
e. P(X>600)
f. P(X<300)
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Homework
Chapter 5: Problems 13, 16 and 30
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CHAPTER SUMMARY
5.1
Probabilities can be calculated using the converse, addition,
and multiplication rules
5.2
The probability distribution is analogous to a frequency
distribution and includes a mean and standard deviation
5.3
The normal curve is a theoretical ideal and therefore cannot be
applied to all distributions
5.4
100% of the data falls under the normal curve, with 50% of the
data falling to either side of the mean
5.5
By converting raw scores to z scores, we can determine the
probability of randomly selecting an individual with that score
from the population
© 2014 by Pearson Higher Education, Inc
Upper Saddle River, New Jersey 07458 • All Rights Reserved