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Chapter 14

MKT 3413 Chapter Notes - Chapter 14: Chi-Squared Distribution, Scatter Plot, Null Hypothesis

Course Code
MKT 3413
Al Burns

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MKT 3413, Alvin Burns
Chapter 14 Textbook Notes: Determining Relationships
Chapter 14 Textbook Notes: Determining Relationships
How Qualtrics Provides the “Total” Package
oRelationship analyses determine whether stable patterns exist
between two (or more) variables
What is a Relationship Between Two Variables?
oEvery scale has unique descriptors, sometimes called levels or labels,
which identify the different positions on that scale
The term levels implies the scale is metric, whereas
The term labels implies that the scale is categorical
oA simple categorical label is “yes” or “no”, for instance, if a respondent
is a buyer (yes) or nonbuyer (no) of a particular product or service
oIf the researcher measured how many times a respondent bought a
product, the level would be the number of times, and the scale would
be metric because this scale would satisfy the assumptions of a real
number scale
oA relationship is a consistent and systematic linkage between the
levels or labels for two variables
A relationship can be used for prediction and it fosters
understanding of the phenomena under study
oA relationship describes the linkage between the levels or labels for
two variables*
Categorical Variables Relationships
oOften graphs such as the stacked cylinder graph, are excellent for
communicating the nature of categorical variable relationships
oStacked cyclinder graphs can be used to show categorical variable
Cross-Tabulation Analysis
oThe analytical technique that assesses the statistical significance of
categorical variable relationships is cross-tabulation analysis
With cross-tabulation, the two categorical variables are
arranged in a cross-tabulation table, defined as the table in
which data are compared using a row-and-column format
oThe intersection of a row and a column is called a cross-tabulation
oA cross-tabulation analysis accounts for all of the relevant label-to-
label relationships and it is the basis for the assessment of statistical
significance of the relationships
oUse a cross-tabulation table for the data defining possible
relationships between two categorical variables*
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oCross-tabulation analysis can be applied to categorical variables with
more than two rows or columns
The intersection cell for the Row Totals column and the
Column Totals row is called the Grand Total
Types of Frequencies and Percentages in a Cross-Tabulation Table
oA frequency table contains the raw counts for the cells based on the
complete data set
A cross-tabulation table contains the raw counts and totals
pertaining to all of the relevant cross-tabulation cells for the
two categorical variables being analyzed
oA frequencies table contains the raw counts of various relationships
possible in a cross-tabulation*
Chi-Square Analysis of a Cross-Tabulation Table
oChi-square (x^2) analysis is the examination of frequencies for two
categorical variables in a cross-tabulation table to determine whether
the variables have a significant relationship
Chi-square analysis begins when the researcher formulates a
statistical null hypothesis that the two variables under
investigation are not related
It is not necessary for the researcher to state this hypothesis in
a formal sense, for chi-square analysis always explicitly takes
this null hypothesis into account
Chi-square begins with the assumption that no
relationship exists between the two categorical
variables under analysis
oChi-square analysis is used to assess the presence of a significant
relationship in a cross-tabulation table*
Observed and Expected Frequencies
oRaw counts referred to as observed frequencies, as they are the
totals observed by counting the number of respondents who are in
each cross-tabulation cell, such as underclass students who did not
attend a movie
oLong ago, someone working with cross-tabulation discovered that if
you multiplied the row total times the column total and divided that
product by the grand total for every crass-tabulation cell, the resulting
expected frequencies would perfectly embody these cell frequencies
if there was no significant relationship present
The expected frequencies perfectly embody the null hypothesis
(of no relationship), so the expected frequencies are a baseline,
and if the observed frequencies are very different from the
expected frequencies, there is reason to believe that a
relationship does exist
oObserved frequencies are found in the sample, whereas expected
frequencies are determined by chi-square analysis procedures*
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Computer Chi-Square Value
oWhen the calculated chi-square value exceeds the critical chi-square
table value, there is a significant relationship between the two
variables under analysis
oBy now, you should realize that when arriving at a computed value, a
statistician will most certainly be comparing it to a table to assess its
statistical significance
oWith higher degrees of freedom, the table chi-square value is larger,
but there is no single value that can be memorized as in our 1.96
number for a normal distribution
A cross-tabulation can have any number of rows and columns,
depending on the labels that identify the various groups in the
two categorical variables being analyzed, and since the degrees
of freedom are based on the number of rows and columns
there is no single critical chi-square value that we can identify
for all cases
Table 14.2- how to determine if you have a significant relationship using chi-
square analysis
oStep 1: Set up the cross-tabulation table and determine the cell counts
known as the observed frequencies
oStep 2: Calculate the expected frequencies using the formula:
Expected cell frequency= cell column total x cell row total /
grand total
oStep 3: Calculate the computed chi-square value using the chi-square
oStep 4: Determine the critical chi-square value from a chi-square
table, using the following formula:
(number of rows – 1) x (number of columns – 1)= degrees of
freedom (df)
oStep 5: Evaluate whether or not the null hypothesis of no relationship
is supported
How to Present a Significant Cross-Tabulation Finding
oCross-tabulation analysis, the best communication vehicle in this case
is a graph
Strongly recommend that you convert your raw counts
(observed frequencies) to percentages for optimal
When you determine that a significant relationship does exist
(that is, there is no support for the null hypothesis of no
relationship), two additional cross-tabulation tables can be
calculated that are valuable in revealing underlying
oThe column percentages table divides the raw frequencies by their
associated column total raw frequency
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