1. Cross-Tabulations

2. 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.

3. Data Types
Prentice-Hall 3

4. 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 4

5. 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

6. 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

7. 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 39

8. Critical Value
What is the critical c2 value if table 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 26

9. c2 Test of Independence Hypotheses & Statistic
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 41

10. 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 42

11. 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 44

12. Expected Frequencies
Prentice-Hall 23

13. Expected Frequencies fe ³ 1 in all cells 132·116 286 132·154 286
132·
154·
Prentice-Hall 45

14. c2 Test of Independence
Prentice-Hall 46

15. c2 Test of Independence Test Statistic: H0: Not Dependent Decision:
H1: Dependent
a = .05
df = (2 - 1)(2 - 1) = 1
Critical Value(s):
Test Statistic:
Decision:
Conclusion:
Reject at a = .05
a = .05
There is evidence of a relationship
Prentice-Hall 47

16. Cross-Tabs
 Please provide the requested information by checking (once) in each category.
  What is your:
 age ____ < 18 ___ ____ > 26
 gender ____ male ____ female
 course load __ < 6 units __ 6 – 12 units __ > 12 units
 gpa __ < 2.0 __ __ __ __ > 3.5
annual income __ < $15k __ $15k - $40k ___ > $40k

17. 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.

18. 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.

19. 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.

20. Cross-Tabs Chi-Square Test ------------------------------------------
Chi-Square Df P-Value

21. Cross-Tabs p-value = 0.8853, Retain H0
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.]

22. 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.

23. Cross-Tabs Chi-Square Test ------------------------------------------
Chi-Square Df P-Value

24. 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.]

25. Cross-Tabs Frequency Table for age by units
Units < > AGE Total
< | | | |
| 17.24% | 20.88% | 33.33% | %
Age | | | |
| 41.38% | 24.18% | % | %
> | | | |
| 41.38% | 54.95% | % | %
UNITS Total
29.00% % % %

26. Questions?

27. ANOVA