Ch 6: Probability
The Role of Probability in Inferential Statistics
probability is used to predict the type of samples that are likely to be
obtained from a population
o probability establishes a connection between samples and populations
o inferential statistics rely on this connection when they use sample data
as the basis for making conclusions about populations
Probability
example: a dice is rolled; there are 6 possible outcomes – 1, 2, 3, 4, 5, 6
o the probability of the dice landing on any 1 value is
example: a coin is tossed; there are two possible outcomes – heads or tails
o the probability of the coin landing on heads is ½ and the probability of
the coin landing on tails is ½
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 the possible outcomes
o if the possible outcomes are identified as A, B, C, D and so on, then:
probability of A is = number of outcomes classified as A / total
number of possible outcomes
o dice roll example: probability of landing on a 2, or p(2) =
all probabilities are between 0 and 1
o 0 < P (A) > 1
o a probability of 0 is a null event & a probability of 1 is a certain event
when population scores are variable, it is impossible to predict with perfect
accuracy which scores will be obtained when you take a sample from the
population
o researchers rely on probability to determine the relative likelihood for
specific examples
o you may not be able to exactly predict outcomes, but you can
determine which outcomes have high and / or low probabilities
Random Sampling
for the definition of “probability” (as it is used here) to be accurate, it is
necessary that the outcomes be obtained by a process called random
sampling
o random sample – requires each individual in the population to have an
equal chance of being selected; a sample obtained by this process is
called a simple random sample
, independent random sample requires that each individual has an equal
chance of being selected and that the probability of being selected stays
constant from one selection to the next if more than one individual is selected
o with replacement; each individual must be returned to population
before making the next selection to keep probabilities from changing
from one selection to the next
o same probability of being selected for the sample, same probability of
being selected first or last
Probability
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
Probability and the Normal Distribution
the normal distribution is symmetrical, with the highest frequency in the
middle and frequencies tapering off as you move toward either extreme
because the locations in the distribution are identified by z-scores, the
percentages shown in the figure apply to any normal distribution regardless
of the values for the mean and the standard deviation
if a vertical line is drawn through a normal distribution, several things occur
o the line divides the distribution into 2 sections; the larger section is
called the body, and the smaller section is called the tail
o the exact location of the line can be specified by a z-score
The Unit Normal Table
the unit normal table lists several different proportions corresponding to each
z-score location
o column A of the table lists z-score values
o for each z-score location, columns B and C list the proportions in the
body and tail, respectively
o 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 (likelihood of a score falling at a particular
point on a distribution)
The Role of Probability in Inferential Statistics
probability is used to predict the type of samples that are likely to be
obtained from a population
o probability establishes a connection between samples and populations
o inferential statistics rely on this connection when they use sample data
as the basis for making conclusions about populations
Probability
example: a dice is rolled; there are 6 possible outcomes – 1, 2, 3, 4, 5, 6
o the probability of the dice landing on any 1 value is
example: a coin is tossed; there are two possible outcomes – heads or tails
o the probability of the coin landing on heads is ½ and the probability of
the coin landing on tails is ½
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 the possible outcomes
o if the possible outcomes are identified as A, B, C, D and so on, then:
probability of A is = number of outcomes classified as A / total
number of possible outcomes
o dice roll example: probability of landing on a 2, or p(2) =
all probabilities are between 0 and 1
o 0 < P (A) > 1
o a probability of 0 is a null event & a probability of 1 is a certain event
when population scores are variable, it is impossible to predict with perfect
accuracy which scores will be obtained when you take a sample from the
population
o researchers rely on probability to determine the relative likelihood for
specific examples
o you may not be able to exactly predict outcomes, but you can
determine which outcomes have high and / or low probabilities
Random Sampling
for the definition of “probability” (as it is used here) to be accurate, it is
necessary that the outcomes be obtained by a process called random
sampling
o random sample – requires each individual in the population to have an
equal chance of being selected; a sample obtained by this process is
called a simple random sample
, independent random sample requires that each individual has an equal
chance of being selected and that the probability of being selected stays
constant from one selection to the next if more than one individual is selected
o with replacement; each individual must be returned to population
before making the next selection to keep probabilities from changing
from one selection to the next
o same probability of being selected for the sample, same probability of
being selected first or last
Probability
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
Probability and the Normal Distribution
the normal distribution is symmetrical, with the highest frequency in the
middle and frequencies tapering off as you move toward either extreme
because the locations in the distribution are identified by z-scores, the
percentages shown in the figure apply to any normal distribution regardless
of the values for the mean and the standard deviation
if a vertical line is drawn through a normal distribution, several things occur
o the line divides the distribution into 2 sections; the larger section is
called the body, and the smaller section is called the tail
o the exact location of the line can be specified by a z-score
The Unit Normal Table
the unit normal table lists several different proportions corresponding to each
z-score location
o column A of the table lists z-score values
o for each z-score location, columns B and C list the proportions in the
body and tail, respectively
o 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 (likelihood of a score falling at a particular
point on a distribution)
