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Cross-Tabulations. Cross-Tabs. The level of measurement used for cross-tabulations are mostly nominal. Even when continuous variables are used (such as age and income), they are converted to categorical variables.

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cross tabs
Cross-Tabs

The level of measurement used for cross-tabulations are mostly nominal. Even when continuous variables are used (such as age and income), they are converted to categorical variables.

When continuous variables are converted to categorical variables, important information (variation) is lost.

data types
Data Types

Prentice-Hall

categorical data
Categorical Data
  • Categorical random variables yield responses that classify
    • Example: Gender (female, male)
  • Measurement reflects number in category
  • Nominal or ordinal scale
    • Examples
      • Did you attend a community college?
      • Do you live on-campus or off-campus?

Prentice-Hall

why concerned about categorical random variables
Why Concerned about Categorical Random Variables?
  • Survey data tends to be categorical … hot/comfortable/cold, sunny/cloudy/fog/rain, yes/no…
  • Know limitations
    • nature of relationship
    • causality
  • Widely used in marketing for decision-making
cross tabs6
Cross-Tabs

The Chi-square, 2, statistic is used to test the null hypothesis.

[Unfortunately, Chi-square, like many other statistics that indicate statistical significance, tells us nothing about the magnitude of the relation.]

Prentice-Hall

c 2 test of independence
c2 Test of Independence
  • Shows whether a relationship exists between two categorical variables
    • One sample is drawn
    • Does not show nature of relationship
    • Does not show causality
  • Used widely in marketing
  • Uses contingency table

Prentice-Hall

critical value
Critical Value

What is the criticalc2 value iftable has 2 rows and 3 columns, a =.05?

If fo = fe, c2 = 0. Do not reject H0

a = .05

df = (2 - 1)(3 - 1) = 2

c2 Table (Portion)

Prentice-Hall

c 2 test of independence hypotheses statistic
c2 Test of Independence Hypotheses & Statistic
  • Hypotheses
    • H0: Variables are not dependent
    • H1: Variables are dependent (related)
  • Test statistic
  • Degrees of freedom: (r - 1)(c - 1)

Observed frequency

Expected frequency

Prentice-Hall

c 2 test of independence expected frequencies
c2 Test of Independence Expected Frequencies
  • Statistical independence means joint probability equals product of marginal probabilities
    • P(A and B) = P(A)·P(B)
  • Compute marginal probabilities
  • Multiply for joint probability
  • Expected frequency is sample size times joint probability

Prentice-Hall

c 2 test of independence an example
c2 Test of Independence An Example

You’re a marketing research analyst. You ask a random sample of 286 consumers if they purchase Diet Pepsi or Diet Coke. At the 0.05 level of significance, is there evidence of a relationship?

Prentice-Hall

expected frequencies13
Expected Frequencies

fe³ 1 in all cells

132·116286

132·154286

132·170286

154·170286

Prentice-Hall

c 2 test of independence15
c2 Test of Independence

Test Statistic:

Decision:

Conclusion:

H0: Not Dependent

H1: Dependent

a = .05

df = (2 - 1)(2 - 1) = 1

Critical Value(s):

Reject at a = .05

a = .05

There is evidence of a relationship

Prentice-Hall

cross tabs16
Cross-Tabs

Please provide the requested information by checking (once) in each category.

  What is your:

  • age ____ < 18 ___ 18 - 26 ____ > 26
  • gender ____ male ____ female
  • course load __ < 6 units __ 6 – 12 units __ > 12 units
  • gpa __ < 2.0 __ 2.0 - 2.5 __ 2.6 - 3.0 __ 3.1 - 3.5 __ > 3.5
  • annual income __ < $15k __ $15k - $40k ___ > $40k
cross tabs17
Cross-Tabs

The information is coded and entered in the file student.sf by letting the first response be recorded as a 1, the second as a 2, etc.

cross tabs18
Cross-Tabs

The hypothesis test generally referred to as

a test of dependence.

The researcher wishes to determine whether the variables are dependent, or, exhibit a relationship.

cross tabs19
Cross-Tabs

Let’s investigate whether a relationship between a student’s gpa and units attempted exists.

H0: GPA and UNITS are not dependent

H1: GPA and UNITS are dependent.

cross tabs20
Cross-Tabs

Chi-Square Test

------------------------------------------

Chi-Square Df P-Value

------------------------------------------

3.67 8 0.8853

------------------------------------------

cross tabs21
Cross-Tabs

p-value = 0.8853, RetainH0

thus, GPA and UNITS are not dependent

[Based on our data, there is no evidence to support the concept that a relationship exists between gpa and units attempted.]

cross tabs22
Cross-Tabs

Let’s investigate whether a relationship between a student’s age and units attempted exist.

H0: AGE and UNITS are not dependent

H1: AGE and UNITS are dependent.

cross tabs23
Cross-Tabs

Chi-Square Test

------------------------------------------

Chi-Square Df P-Value

------------------------------------------

9.89 4 0.0423

------------------------------------------

cross tabs24
Cross-Tabs

p-value = 0.0423, Reject H0

thus, AGE and UNITS are dependent

[Based on our data, there is sufficient evidence to support the concept that a relationship exists between age and units attempted.]

cross tabs25
Cross-Tabs

Frequency Table for age by units

Units<6 6-12 >12 AGE Total

--------------------------------------------------------

<18 | 10 | 19 | 17 | 46

| 17.24% | 20.88% | 33.33% | 23.00%

--------------------------------------------------------

Age18-26 | 24 | 22 | 16 | 62

| 41.38% | 24.18% | 31.37% | 31.00%

--------------------------------------------------------

>26 | 24 | 50 | 18 | 92

| 41.38% | 54.95% | 35.29% | 46.00%

--------------------------------------------------------

UNITS Total 58 91 51 200

29.00% 45.50% 25.50% 100.00%

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