### Chapter 6: Probability

```Chapter 6: Probability
1
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.
2
Probability (cont.)
• Whenever the scores in a population are
variable it is impossible to predict with perfect
accuracy exactly which score or scores 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.
4
Probability (cont.)
• Probability is determined by a fraction or
proportion.
• When a population of scores is
represented by a frequency distribution,
probabilities can be defined by proportions
of the distribution.
• In graphs, probability can be defined as a
proportion of area under the curve.
5
Probability and the Normal
Distribution
• If a vertical line is drawn through a normal
distribution, several things occur.
1. The exact location of the line can be
specified by a z-score.
2. The line divides the distribution into
two sections. The larger section is
called the body and the smaller section
is called the tail.
6
Probability and the Normal
Distribution (cont.)
• 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.
7
Probability and the Normal
Distribution (cont.)
• To find the probability corresponding to a
particular score (X value), you first transform the
score into a z-score, then look up the z-score 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.
9
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.
10
Probability and the Binomial
Distribution
• Binomial distributions are formed by a series of
observations (for example, 100 coin tosses) for
which there are exactly two possible outcomes
• The two outcomes are identified as A and B, with
probabilities of p(A) = p and p(B) = q.
• 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.
12
Probability and the Binomial
Distribution (cont.)
• When pn and qn are both 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.
13
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.
15
Probability and Inferential Statistics (cont.)
• The general goal of inferential statistics is to use
the information from a sample to reach a general
population.
• Typically a researcher begins with a sample.
• 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.
16
```