practice 5 cj stats
ismails95
5ProbabilityandtheNormalCurve.ppt© 2014 by Pearson Higher Education, Inc Upper Saddle River, New Jersey 07458 • All Rights Reserved
Fox/Levin/Forde, Elementary Statistics in Criminal Justice Research, 4e
Chapter 5: Probability and the Normal Curve
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© 2014 by Pearson Higher Education, Inc Upper Saddle River, New Jersey 07458 • All Rights Reserved
Calculate probabilities and understand the rules of probability
Understand the concept of a probability distribution
List the characteristics of the normal curve
Understand the area under the normal curve
Calculate and use z scores
CHAPTER OBJECTIVES
5.1
5.2
5.3
5.4
5.5
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Calculate probabilities and understand the rules of probability
Learning Objectives
After this lecture, you should be able to complete the following Learning Outcomes
5.1
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- Until now, we have drawn conclusions about data that we have observed
- cornerstone of decision making is probability
- probability – refers to the relative likelihood of occurrence of any given outcome or event
- number of times that event can occur relative to the total number of times any event can occur
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- P varies from 0 to 1
- often expressed as a decimal
- probabilities near 0 (.05 or .1) imply very unlikely occurrences
- probabilities near 1 (.90, .95 or .99) imply very probable or likely outcomes
- a probability of 1 constitutes certainty
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Probability
5.1
P = 1
The outcome is certain
P = .5
The outcome is as likely to happen as not
happen
P = 0
The outcome is
impossible
The relative likelihood of occurrence of any given outcome
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The Rules of Probability
5.1
Probability
Converse Rule: The probability
that something will not occur
Addition Rule: The probability of obtaining
one of several different and distinct outcomes
Multiplication Rule: The probability of
obtaining two or more outcomes in combination
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Probability: Example
5.1
Heads or Tails?
Probability: Example
5.1
Heads or Tails?
Understand the concept of a probability distribution
Learning Objectives
After this lecture, you should be able to complete the following Learning Outcomes
5.2
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Probability Distributions
5.2
Mean = μ
Standard Deviation = σ
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Directly analogous to a frequency distribution
- Except it is based on probability theory
- probabilities sum to 1
- can be plotted like frequency distributions
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Difference bt Probability and Frequency Distributions
- probability distribution is based on probability theory
- describes what SHOULD happen
- frequency distribution describes what DID happen
- probability distribution is essentially a frequency distribution for an infinite number of flips
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Mean and SD of a Probability Distribution
- probability distribution has a mean represented by the Greek letter mu μ
- SD is represented by sigma σ
- variance is represented by σ2
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Comparing Notation
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| Frequency Distribution | Probability Distribution | |
| Mean | x̄ (X bar) | μ (mu) |
| Variance | s2 | σ2 |
| Standard Deviation | s | σ (sigma) |
List the characteristics of the normal curve
Learning Objectives
After this lecture, you should be able to complete the following Learning Outcomes
5.3
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The Normal Curve
- normal curve – a theoretical or ideal model
- obtained from a mathematical equation
- can be used for
describing distributions of scores
interpreting the SD
making statements of probability
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5.3
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- Smooth
- Symmetrical
- Unimodal
- Mean = Median = Mode
- Infinite in both directions
- Probability distribution
- Mean = μ; Standard Deviation = σ
- Areas under the normal curve = 100%
Characteristics of the Normal Curve
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5.3
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The normal curve is a theoretical ideal.
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 influence the validity of our conclusions.
The Reality of the Normal Curve
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Understand the area under the normal curve
Learning Objectives
After this lecture, you should be able to complete the following Learning Outcomes
5.4
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© 2014 by Pearson Higher Education, Inc Upper Saddle River, New Jersey 07458 • All Rights Reserved
5.4
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© 2014 by Pearson Higher Education, Inc Upper Saddle River, New Jersey 07458 • All Rights Reserved
5.4
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© 2014 by Pearson Higher Education, Inc Upper Saddle River, New Jersey 07458 • All Rights Reserved
5.4
*
© 2014 by Pearson Higher Education, Inc Upper Saddle River, New Jersey 07458 • All Rights Reserved
5.4
*
Calculate and use z scores
Learning Objectives
After this lecture, you should be able to complete the following Learning Outcomes
5.5
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Standard Scores and the Normal Curve
- How do we determine the sigma distance of any given raw score?
- IE How do we translate our raw score?
- We can translate any given raw score into sigma units by dividing the distance of the raw score from the mean of the SD
- this yields a value called a z score
- z score – indicates the strength and degree that any given raw score deviates from the mean of a distribution on a scale of sigma units
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5.5
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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
Standard Scores and the Normal Curve
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5.5
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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.
Finding Probability under the Normal Curve
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Step by Step – Probability Under the Normal Curve
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*
*
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Step by Step – Finding Scores from Probability Based on the Normal Curve
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© 2014 by Pearson Higher Education, Inc Upper Saddle River, New Jersey 07458 • All Rights Reserved
Probabilities can be calculated using the converse, addition, and multiplication rules.
The probability distribution is analogous to a frequency distribution and includes a mean and standard deviation.
The normal curve is a theoretical ideal and therefore cannot be applied to all distributions.
100% of the data falls under the normal curve, with 50% of the data falling to either side of the mean.
By converting raw scores to z scores we can determine the probability of randomly selecting an individual with that score from the population.
CHAPTER SUMMARY
5.1
5.2
5.3
5.4
5.5
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Number of times the outcome or event can
occur
(F)
Total number of times any outcome or eve
nt can occur
P
=
(
)
(
)
F1F
PP
=-
(
)
(
)
(
)
A or BAB
PPP
=+
(
)
(
)
(
)
A and BA X B
PPP
=
1
.5
2
=
1
.5
2
=
(
)
(
)
.5.5.25
=
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
HTTHHTTH
.50.50.50.50
.25.25
.50
PPPPPP
+=+
=+
=+
=
X
z
m
s
-
=
mean of a distribution
standard deviation of a distribution
standard score
z
m
s
=
=
=
Xz
ms
=+
