Chapter 6
Probability
Introduction
We usually start a study asking questions
about the population.
But we conduct the research using a
sample.
The role of inferential statistics is to use
the sample data as the basis for
answering questions about the population.
Introduction (cont.)
To accomplish this goal, inferential
procedures are typically built around the
concept of probability.
Specifically, the relationships between
samples and populations are usually
defined in terms of probability.
By knowing the makeup of a population,
we can determine the probability of
obtaining specific samples.
This way, probability gives us a connection
between populations and samples.
This way, probability gives us a connection
between populations and samples which
will be the foundation for inferential
statistics (later chapters)
The marble samples began with a
population and ended with a sample
obtained.
The goal of inferential statistics begins
with a sample that answers general
questions about the population.
Two steps to reach the goal:
– Develop probability as a bridge from
population to samples
– Then, reverse the probability rules to allo the
samples to move to populations.
Figure 6.1
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Wadsworth Group, a division of Thomson Learning
The role of probability in inferential statistics
The role of probability in inferential statistics. Probability is used to
predict what kind of samples are likely to be obtained from a population.
Thus, probability establishes a connection between samples and
populations. Inferential statistics rely on this connection when they use
sample data as the basis for making conclusions about populations.
Probability Definition
In a situation where several different
outcomes are possible, we define the
probability for any particular outcome as a
fraction or proportion. If the possible
outcomes are identified as A, B, C, D, and
so on, then
Probability of A = number of outcomes classified as A
Total number of possible outcomes
Example
A deck of cards – there are 52 cards.
The probability of choosing the king of
hearts is p = 1/52
The probability of choosing an ace is p =
4/52 = 1/13 = .0769
Use a notation system
– p(hearts)
– p(aces)
Note:
Probability is defined as a proportion.
Can restate any probability problem as a
proportion problem.
– What is the probability of obtaining a king
from a deck of cards? 4/52
– Out of the whole deck, what proportion are
kings? 4/52
There is a reason to understand this now.
Example:
p(tails) = ½ = .50 = 50%
Any of the three forms are acceptable.
Question:
– If you had a jar of all white marbles, what is
the possibility of choosing a black marble?
– What is the probability of choosing a white
marble?
Random Sampling
For the definition of probability to be
accurate, the outcomes must be obtained
through random sampling:
Random sampling must satisfy two
requirements:
– Each individual in the population must have
an equal chance of being selected
Assures no bias in the selection process
Requirements for Random Sample (cont.)
– If more than one individual is to be selected
for the sample, there must be constant
probability for each and every selection
p(jack of diamonds) = 1/52 for the first draw
p(jack of diamonds) = 1/51 for the second draw
p(jack of diamonds) = 0 if the jack of diamonds
was the first draw
This contradicts the first requirement that
states the probability must stay constant.
Sampling with replacement
To keep the probabilities from changing
from one selection to the next, it is
necessary to replace each sample before
you make the next selection
– Sampling with replacement
Types of random sampling
Simple random sample
Independent random sample
Sampling with replacement
Sampling without replacement
There are different sampling techniques
used by researchers
Probability and Frequency Distribution
In education, we are usually concerned
with probability that will involve a
population of scores that can be displayed
in a frequency distribution graph.
If the graph represents the entire
population, then a portion of the graph
represents a different portion of the
population.
Probability and Frequency Distribution (cont.)
Because probability and proportion are
equivalent, a particular proportion of the
graph corresponds to a particular
probability in the population.
Thus, whenever a population is presented
in a frequency distribution graph, it will be
possible to represent probabilities as
proportions of the graph.
Example:
N = 10 scores
1, 1, 2, 3, 3, 4, 4, 4, 5, 6
If you take a random sample of n=1 score
from this population, what is the probability
of obtaining a score greater than 4?
p(X > 4) = ?
p(X > 4) = ?
Using this criteria, there are 2 scores that meet this
criterion out of the total group of N=10 scores, so
p = 2/10
We are now defining probability as the proportion of area
in the frequency distribution graphs.
– very graphic and concrete way of representing probability
What is the probability of selecting a score
less than 5?
p(X < 5) = ?
What part of the graph is unshaded?
p = 8/10
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Figure 6.3
The normal distribution
Probability and the Normal Distribution
Note that the normal distribution is
symmetrical
Highest frequency in the middle
Frequencies tapering off as you move
towards the extremes
Normal shape can also be described by
the proportions of area contained in each
section of the distribution
Probability and the Normal Distribution (cont.)
Statisticians often identify sections of a
normal distribution by using z-scores
Remember that z-scores measure
positions in a distribution in terms of
standard deviations from the mean
The graph shows the percentage of scores
that fall in each of these sections
Figure 6.4
The normal distribution following a z-score transformation
Copyright © 2002 Wadsworth Group. Wadsworth is an imprint of the
Wadsworth Group, a division of Thomson Learning
Of the population
Of the population
Of the population
In this way it is possible to define a normal
distribution in terms of its proportions
That is, a distribution is normal if and only
if it has all the right proportions
Note: Because the normal distribution is
symmetrical the sections on the left side of
the distribution have exactly the same
proportions as the corresponding sections
on the right side
Note: Because the locations in the
distribution are identified by z-scores, the
proportions shown in the figure apply to
any normal distribution regardless of the
values for the mean and the standard
deviation
When any distribution is transformed into
z-scores, the mean becomes zero and the
standard deviation becomes one
The process of answering probability
questions about a normal distribution
Example 6.2
– Adult heights form a normal distribution with a
mean of 68 inches and a standard deviation
of 6 inches.
– Given this information about the population
and the known proportions for a normal
distribution
– We can determine the probability associated
with specific examples
Figure 6.4
The normal distribution following a z-score transformation
Copyright © 2002 Wadsworth Group. Wadsworth is an imprint of the
Wadsworth Group, a division of Thomson Learning
Of the population
Of the population
Of the population
For example…
What is the probability of randomly
selecting an individual from this
population who is taller than 6 feet 8
inches (X=80 inches)
P(X > 80)
1. The probability question is translated into
a proportion question: Out of all possible
adult heights, what proportion is greater
than 80?
2. We know that “all possible adult heights”
is simply the population distribution.
The m = 68, so the score X = 80 it to the right
of the mean.
Because we are interested in all heights
greater than 80, we shade in the area to the
right of the 80.
This area represents the proportion we are
trying to determine.
Identify the exact position of X=80 by
computing a z-score. For this example,
z = X - m = 80 - 68 =
s
6
12 = 2.00
6
A height of 80 is 2 s.d. above the mean
and corresponds to a z-score of +2.00.
The proportion we are trying to determine
may now be expressed in terms of a zscore
p(X > 80) = ?
p(z > 2.00) = ?
p(X > 80) = p(z > 2.00) = 2.28%
Figure 6.4
Copyright © 2002 Wadsworth Group. Wadsworth is an imprint of the
Wadsworth Group, a division of Thomson Learning
The normal distribution following a z-score transformation
All normal distributions
will have 2.28% of the
scores in the tail
beyond z = +2.00
Of the population
Unit Normal Table
The graph of the normal distribution shows
proportions for only a few selected z-score
values.
A more complete listing of z-scores and
proportions is provided in the unit normal
table.
This table lists proportions of the normal
distribution for a full range of possible zscore values.
Copyright © 2002 Wadsworth Group. Wadsworth is an imprint of the
Wadsworth Group, a division of Thomson Learning
Figure 6.6
A portion of the unit normal table
z= 0.25
59.87%
40.13%
Column A lists z-score values
corresponding to different locations in a
normal distribution
Column B and C: identify the proportion of
the distribution in each of the two sections
Column B: presents the proportion in the
body (the larger portion)
Column C: presents the proportion in the
tail
When you use the unit table…
Keep in mind…
– The body corresponds to the larger part
(either right-hand or left-hand)
– The tail corresponds to the smaller part of
the distribution
– The proportions on the right-hand side are
exactly the same as the corresponding
proportions on the left-hand side
When you use the unit table…
Proportions will always be positive (even if
z-score is negative
For any specific z-score value, the two
proportions will always add up to 1.00 (the
whole distribution)
Let’s review…
The unit normal table lists relationships between
z-scores locations and proportions in a normal
distribution
For any z-score location, you can use the table
to look up the corresponding proportions
If you know the proportions, you can use the
table to look up the specific z-score location
Because we have defined probability as
equivalent to proportion, you can also use the
unit normal table to look up probabilities for
normal distributions
Example 6.3A
15.87%
What proportion of the normal distribution corresponds to z-score
values greater than z = 1.00?
Shade the area you are trying to determine.
Look up z=1.00 in Column A
Read Column C for the proportion which is 0.1587 or 15.87%
Example 6.3B
For a normal distribution, what is the probability of selecting a z-score
less than z = 1.50?
93.32%
Example 6.3C
What is the proportion of the normal distribution that corresponds
to the tail beyond z = - 0.50?
30.85%
If you have the proportion, can you find the
z-score?
For a normal distribution, what z=score separates the top 10% from the
remainder of the distribution?
10% = .1000 Locate .1000 on the table – Column C or
90% = .9000 Locate .9000 on the table – Column B
Choose the closest number that you can. For this case, it would be 0.1003 in
Column C.
Z= +1.28 (Make sure to designate + or -).
Example 6.4B
Body
Tail
For a normal distribution, what z-score value forms the
boundary between the top 60% and the bottom 40% of
the scores?
Column B - 0.6000
Column C - 0.4000
z = - 0.25
Probabilities, Proportions, and
Scores (X values)
In most situations, it will be necessary to
find probabilities for specific X values
– Transform the X value into z-scores
– Use the unit normal table to look up the
proportions corresponding to the z-score
values
Example
It is known that IQ scores form a normal
distribution with m = 100 and s = 15.
What is the probability of randomly
selecting an individual with an IQ score
greater than 130?
Example 6.5
P (X > 130) = ?
We want to find the proportion of the IQ distribution that corresponds
to scores greater than 130.
Change the X values into z-scores
p (X > 130) = ?
X = 130
z = X – m = 130 – 100 = 30 = 2.00
s
15
15
Look up the z-score value in the unit
normal table
p (X>130) = 0.0228 = 2.28%
Finding proportions/probabilities
located between two scores
This example demonstrates the process of
finding the probability of selecting a score
that is located between two specific
values.
We are now looking for a proportion
defined by a slice from the middle of the
normal distribution.
Finding proportions/probabilities
located between two scores
The final answer does not correspond to
either the body or the tail of the
distribution, which means that you cannot
read the answer directly from the table.
Instead, you must use the information in
the table to calculate the final answer.
Example 6.6
The distribution of SAT scores in normal
with m = 500 and s = 100.
What is the probability of randomly
selecting an individual with a score
between X = 600 and X = 700?
In other words…
Find p (600 < X < 700) = ?
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Wadsworth Group, a division of Thomson Learning
Figure 6.10
The distribution for Example 6.6
Transform each X value into a z-score
For X = 600
z = X – m = 600 – 500 = 100 = 1.00
s
100
100
For X = 700
z = X – m = 700 – 500 = 200 = 2.00
s
100
100
Now find the proportion of the normal
distribution that is located between z =
+1.00 and z = +2.00
We can approach the problem one of two
ways…
Approach 1
This method focuses on the proportions in
the tail of the distribution.
Use Column C to find the proportion in the
tail beyond z = +1.00
0.1587
This includes the shaded portion that we
are trying to find, but it also includes an
extra portion in the tail beyond z = +2.00
Use the table again to find the extra
portion beyond z=+2.00
z = +2.00 = 0.0228
Now subtract the two
p(600 < X < 700) =
0.1587 – 0.0228 =
0.1359 =
13.59%
Approach 2
Find how much of the distribution is
located outside the section we want to
measure.
We want the unshaded areas of the
distribution
z = 2.00 = 0.0228
z = 1.00 = 0.8413
The total area (that we do not want) is
0.0228 + 0.8413 =
.8641
Subtract from 1.000 (because the whole
distribution is 1.000)
1.0000 - .8641 =
.1359 =
13.59%
Finding scores corresponding to specific
proportions or probabilities
In the previous examples, the problem
was to find the proportion or probability
corresponding to specific X values.
The two-step process is illustrated in the
following example:
Figure 6.11
A map for probability problems
We have only described how to go
clockwise in this process.
Copyright © 2002 Wadsworth Group. Wadsworth is an imprint of the
Wadsworth Group, a division of Thomson Learning
Start here
We can go backwards
to find a
corresponding score
for a certain
proportion.
Example 6.7
Scores on the SAT form a normal
distribution with m = 500 and s = 100.
What is the minimum score necessary to
be in the top 15% of the SAT distribution?
Begin with 15% = .1500
We are looking for a score.
Figure 6.11
A map for probability problems
We have only described how to go
clockwise in this process.
Copyright © 2002 Wadsworth Group. Wadsworth is an imprint of the
Wadsworth Group, a division of Thomson Learning
Start here
We can go backwards
to find a
corresponding score
for a certain
proportion.
We can go from proportion to X by going
via z=scores.
1. Use the unit normal table to find the z-
score that corresponds to a proportion of
0.15
2. Look at the graphic
We will need to use Column C because the shaded area is the tail.
The closest value in the table is 0.1492, and the z-score that
corresponds to this proportion is z = 1.04.
Next: Determine whether the z-score is
positive or negative.
In this case z = + 1.04
Now to change the z-score into an X value
– Use the z-score equation:
X = m + zs
= 500 + 1.04 (100)
= 500 + 104
= 604