Download Chapter 6: Probability

COURSE: JUST 3900
INTRODUCTORY STATISTICS
FOR CRIMINAL JUSTICE
Chapter 6:
Probability
Instructor:
Dr. John J. Kerbs, Associate Professor
Joint Ph.D. in Social Work and Sociology
Probability
Probability is a method for measuring and
quantifying the likelihood of obtaining a
specific sample from a specific population.
 We define probability as a fraction or a
proportion.
 The probability of any specific outcome is
determined by a ratio comparing the
frequency of occurrence for that outcome
relative to the total number of possible
outcomes.

Probability (Continued)
Inferential statistics use sample
data to answer questions about
populations
Probability is used to predict
what kinds of samples are likely
to be obtained from a
population.
Probability (Continued)

Whenever the scores in a population are variable, it
is impossible to predict with perfect accuracy exactly
which score(s) will be obtained when you take a
sample from the population.

In this situation, researchers rely on probability to
determine the relative likelihood for specific samples.

Thus, although you may not be able to predict exactly
which value(s) will be obtained for a sample, it is possible
to determine which outcomes have high probability and
which have low probability.
Probability (Continued)

Definition: For a situation in which several different outcomes
are possible, the probability for any specific outcome is
defined as a fraction or a proportion of all possible outcomes.
If the possible outcomes are identified as A, B, C, D, and so
on, then the following formula applies:

Other Notation: p(outcome) = . . .
Remember: You can express probabilities as
 1.
Fractions
 2.
Percentages
 3.
Decimals - - most commonly used approach

Probability and Sampling

To assure that the definition of probability is
accurate, the use of random sampling is
necessary.

Random sampling requires that each member of
a population has an equal chance of being
selected.
 If
more than one person is being selected, the
probabilities must stay constant from one selection to
the next - - Thus, the next requirement below.

Independent random sampling includes the
conditions of random sampling and further requires
that the probability of being selected remains
constant for each selection
Probability and Sampling

Two main approaches to random sampling

Approach #1: Keeps probabilities from changing
from one selection to another
 Random
Sampling with Replacement: This approach to
sampling requires that you return each person to the
population before you make another selection.
aka - - Independent Random Sampling, which keeps a
constant probability of selection

Approach #2: Selections are random, but there is
no maintenance of constant probability.
 Random
Sampling without Replacement: This approach
does not maintain constant probability from one
selection to the next selection
Probability and Sampling

Card example for sampling with replacement
with constant probability
p(ace of diamonds) = 1/52, replace card
 p(ace of diamonds) = 1/52, replace card…


Card example for sampling without
replacement and no assumption of constant
probability
p(ace of diamonds) = 1/52, then next draw
 p(ace of diamonds) = 1/51, then next draw
 p(ace of diamonds) = 1/50, then next draw

 NOTICE
HOW PROBABILITY OF SELECTION IS NOT
CONSTANT WITHOUT REPLACEMENT
Probability (Continued)
When a population of scores is
represented by a frequency distribution,
probabilities can be defined by proportions
of the distribution.
 Probability values are expressed by a
fraction or proportion.
 In graphs, probability can be defined as a
proportion of area under the curve.

Probability and the Normal
Distribution
Probability and the Normal
Distribution

If a vertical line is drawn through a normal
distribution, several things occur.
The line divides the distribution into two
sections. The larger section is called the body
and the smaller section is called the tail.
 The exact location of the line can be specified
by a z-score.

Probability and the Normal
Distribution (Continued)

The unit normal table lists several
different proportions corresponding to
each z-score location.




Column A of the table lists z-score values.
For each z-score location, columns B and C list the
proportions in the body and tail, respectively.
Finally, column D lists the proportion between the
mean and the z-score location.
Because probability is equivalent to
proportion, the table values can also be
used to determine probabilities.
The Normal Distribution Example
z = 0.34
B + C = 1.00
Probability and the Normal
Distribution (cont'd.)
To find the probability corresponding to a
particular score (X value), you first transform
the score into a z-score, then look up the zscore in the table and read across the row to
find the appropriate proportion/probability.
 To find the score (X value) corresponding to a
particular proportion, you first look up the
proportion in the table, read across the row to
find the corresponding z-score, and then
transform the z-score into an X value.

Probability and the Normal
Distribution (cont'd.)

4 Key Facts about Unit Normal (z) Tables
 1.
The body (column B) always represents the larger
part of the distribution and the tail (column C) is always
the smaller section, whether on the right or left side.
 2. The normal distribution is symmetrical; therefore, the
proportions will be the same for the positive and
negative values of a specific z-score .
 3. Proportions are always positive, even if the
corresponding z-score is negative. To find proportions
for negative z-scores, look up the corresponding
proportion for the positive value of z.
 4. A negative z-score means that the tail of the
distribution is on the left side and the body is on the
right, and vice versa for a positive z-score.
Probability and the Normal
Distribution (cont'd.)
See Column C in Unit
Normal Table on page 699
Note: z = + 0.25 here.
Your book had a typo
on p. 174 and listed it
as a negative z-score.
Please correct in your
book
Note: z = - 0.25 here.
Book is correct for
this side of the graph
Percentiles and Percentile Ranks
The percentile rank for a specific X value is
the percentage of individuals with scores at or
below that value.
 When a score is referred to by its rank, the
score is called a percentile. The percentile
rank for a score in a normal distribution is
simply the proportion to the left of the score.

Percentiles and Percentile Ranks
0 +0.25
P(z>1.00)
Tail = 0.1587
Or 15.87%
Look up 10% in Tail
Tail = 0.1003
For z = 1.28
P(z<1.50)
Body = 0.9332
Or 93.32%
P(z<-0.50)
Tail = 0.3085
Or 30.85%
Think Symmetry!!!!!
30% on each side of μ = 0
Look up z for .30 or 30%
between μ=0 and z
(see Column D)
Z=+/- 0.84 for proportion .2995
Percentiles and Percentile Ranks
Percentile ranks represent specific scores as a
percentage of individuals in the distribution who
have scores that are less than or equal to the
specific score.
 For example, if 80% of all JUST3900 students
had term grades that were less than or equal to
87, then a score of 87 has a percentile rank of
80%.
 Thus, a score of 87 puts students at the
80th percentile.

Percentiles and Percentile Ranks

Imagine that the population of all drug abusing
offenders are assessed for their drug cravings
on a scale from 0 (no cravings) to 150 (intense
cravings).

The assessment finds μ = 100 and σ = 15 and
we need to determine what proportion of drug
abusing offenders have cravings that fall
between scores of 115 and 130.
Percentiles and Percentile Ranks
The assessment finds μ = 100 and σ = 15 and
we need to determine what proportion of drug
abusing offenders have cravings that fall
between scores of 115 and 130.
 Step 1: Find z-scores for two values




For x = 115: z = (x- μ)/ σ = (115-100)/15 = 1.00
For x = 140: z = (x- μ)/ σ = (130-100)/15 = 2.00
Step 2: Find corresponding proportion between
the two z-scores

p(1.00<z<2.00) = .1587 - .0228 = 0.1359 or 13.59%
Percentiles and Percentile Ranks

Please note that you must be able to convert
 raw scores (i.e., x values) into z-scores

Use the z-score formula z = (x- μ) / σ
z-scores into proportions and probabilities
 proportions and probabilities into z-scores
 Z-scores into raw scores (i.e., x values)



Use the z-score formula z = (x- μ) / σ
Please remember that it is impossible to directly
transform an x value into a proportion or
probability without first converting the x value
into a z-score and then into a probability or
proportion
Probability and the Binomial
Distribution for Two Outcomes
Binomial distributions are formed by a series
of observations (for example, 100 coin tosses)
for which there are exactly two possible
outcomes (heads and tails)
 The two outcomes are identified as A and B,
with probabilities of p(A) = p and p(B) = q.

p

+ q = 1.00
The distribution shows the probability for each
value of X, where X is the number of
occurrences of A in a series of n observations.
Probability and the Binomial
Distribution (cont'd.)



When pn and qn are both equal to or greater than 10, the
binomial distribution is closely approximated by a normal
distribution with a mean of μ = pn and a standard deviation of
σ = npq.
In this situation, a z-score can be computed for each value of
X and the unit normal table can be used to determine
probabilities for specific outcomes.
Within the normal distribution, each value of X has a
corresponding z-score as follows:
Binomial Distributions
Probability and Inferential
Statistics
Probability is important because it establishes
a link between samples and populations.
 For any known population, it is possible to
determine the probability of obtaining any
specific sample.
 In later chapters, we will use this link as the
foundation for inferential statistics.

Probability and Inferential
Statistics (cont'd.)
The general goal of inferential statistics is to
use the information from a sample to reach a
general conclusion (inference) about an
unknown population.
 Typically a researcher begins with a sample.

Probability and Inferential
Statistics (cont'd.)
If the sample has a high probability of being
obtained from a specific population, then the
researcher can conclude that the sample is
likely to have come from that population.
 If the sample has a very low probability of
being obtained from a specific population,
then it is reasonable for the researcher to
conclude that the specific population is
probably not the source for the sample.

Research Study - - Likelihood of Predicting a Card’s Suit 15
Times in a Row for in 48 Trials
Actual Score Value
Real Limits
μ = pn
μ = (1/4)*48 = 12
qn = (3/4)*48
Thus, z = 1.17 and
p = .1210 or 12.10%
Research Study
If the x-score is
440, is this an
extreme value
as defined by
the book?
The goal of this study is to determine whether
the treatment has an effect.
As a primer for the next chapter, extreme
effects are considered those that are defined
by scores that are very unlikely to be obtained
from the original population by random
chance, thus providing evidence of treatment
effects.
Cutoff scores for 1-tail tests:
Scores with p<0.05 (z = 1.65)
Scores with p<0.01 (z = 2.33)
Scores with p<0.001 (z = 3.11)
Value as discussed in
book on page 190-191
Cutoff scores for 2-tail tests:
Scores with p<0.05 (z = +/-1.96) w/0.025 per tail
Scores with p<0.01 (z = +/-2.58) w/0.005 per tail
Scores with p<0.001 (z = +/-3.30) w/0.0005 per
tail