1. S519: Evaluation of Information Systems
Social Statistics
Inferential Statistics
Chapter 15: Chi-square

2. Last week
Linear regression
Slope
Intercept

3. This week
What is chi-square
CHIDIST
Non-parameteric statistics

4. Parametric statistics
A main branch of statistics
Assuming data with a type of probability distribution (e.g. normal distribution)
Making inferences about the parameters of the distribution (e.g. sample size, factors in the test)
Assumption: the sample is large enough to represent the population (e.g. sample size around 30).
They are not distribution-free (they require a probability distribution)

5. Nonparametric statistics
Nonparametric statistics (distribution-free statistics)
Do not rely on assumptions that the data are drawn from a given probability distribution (data model is not specified).
It was widely used for studying populations that take on a ranked order (e.g. movie reviews from one to four stars, opinions about hotel ranking). Fits for ordinal data.
It makes less assumption. Therefore it can be applied in situations where less is known about the application.
It might require to draw conclusion on a larger sample size with the same degree of confidence comparing with parametric statistics.

6. Nonparametric statistics
Nonparametric statistics (distribution-free statistics)
Data with frequencies or percentage
Number of kids in difference grades
The percentage of people receiving social security

7. One-sample chi-square
One-sample chi-square includes only one dimension
Whether the number of respondents is equally distributed across all levels of education.
Whether the voting for the school voucher has a pattern of preference.
Two-sample chi-square includes two dimensions
Whether preference for the school voucher is independent of political party affiliation and gender

8. Compute chi-square One-sample chi-square test
O: the observed frequency
E: the expected frequency

9. Example
Question: Whether the number of respondents is equally distributed across all opinions
One-sample chi-square
Preference for School Voucher 
for
maybe
against
total 23 17 50 90

10. Chi-square steps Step1: a statement of null and research hypothesis
There is no difference in the frequency or proportion in each category
There is difference in the frequency or proportion in each category

11. Chi-square steps
Step2: setting the level of risk (or the level of significance or Type I error) associated with the null hypothesis
0.05

12. Chi-square steps Step3: selection of proper test statistic
Frequencynonparametric procedureschi-square

13. Chi-square steps
Step4. Computation of the test statistic value (called the obtained value)
category
observed frequency (O)
expected frequency (E)
D(difference)
(O-E)2
(O-E)2/E
for 23 30 7 49
1.63
maybe 17 13
169
5.63
against 50 20
400
13.33
Total 90
20.60

14. Chi-square steps
Step5: Determination of the value needed for rejection of the null hypothesis using the appropriate table of critical values for the particular statistic
Table B5
df=r-1 (r= number of categories)
If the obtained value > the critical value  reject the null hypothesis
If the obtained value < the critical value  accept the null hypothesis

15. Chi-square steps
Step6: a comparison of the obtained value and the critical value is made
20.6 and 5.99

16. Chi-square steps Step 7 and 8: decision time
What is your conclusion, why and how to interpret?

17. Another example
We’ll settle the age-old debate of whether people can actually detect their favorite cola based solely on taste. For 30 coke-lovers, I blindfold them, and have them sample 3 colas…is there a true difference, or are these preference differences explainable by chance?

18. Hypothesis
Null: There are no preferences: The population is divided evenly among the brands
Alternate: There are preferences: The population is not divided evenly among the brands

19. Chance Model df = C -1 = 3 -1 = 2, set α = .05
For df = 2, X2-crit = 5.99

20. Calculate Chi-Square category observed frequency (O)
expected frequency (E)
D(difference)
(O-E)2
(O-E)2/E
Coke 13 10 3 9
0.9
Pepsi 1
0.1
RC Cola 8 2 4
0.4
Total 30
1.4

21. Decision and Conclusion
Conclude that the preferences are evenly divided among the colas when the logos are removed.

22. Excel functions CHIDIST (x, degree of freedom) CHIDIST(20.6,2)
>0.05

23. More non parametric statistics
Table 15.1 (P297)