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

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**preview**shows pages 1-3. to view the full**12 pages of the document.**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

relationships*

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

cell

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

formula

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

communication

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

relationships

oThe column percentages table divides the raw frequencies by their

associated column total raw frequency

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