1. Chapter 16 The Chi-Square Statistic
PowerPoint Lecture Slides Essentials of Statistics for the Behavioral Sciences Seventh Edition by Frederick J. Gravetter and Larry B. Wallnau

2. Chapter 16 Learning Outcomes
Explain when chi-square test is appropriate 2
Test hypothesis about shape of distribution using chi-square goodness of fit 3
Test hypothesis about relationship of variables using chi-square test of independence 4
Evaluate effect size using phi-coefficient or Cramer’s V

3. Concepts to review Proportions (math review, Appendix A)
Frequency distributions (Chapter 2)

4. 16.1 Parametric and nonparametric statistical tests
Hypothesis tests used thus far tested hypotheses about population parameters
Parametric tests share several assumptions
Normal distribution in the population
Homogeneity of variance in the population
Numerical score for each individual
Nonparametric tests are needed when the research situation does not conform to the requirements of parametric tests.

5. Chi-Square and other nonparametric tests
Do not state the hypotheses in terms of a specific population parameter
Make few assumptions about the population distribution
Often termed distribution free tests
Participants usually classified into categories
Nominal or ordinal scales are used
Data for nonparametric tests are frequencies

6. 16.2 Chi-Square Test for Goodness of Fit
Uses sample data to test hypotheses about the shape or proportions of a population distribution.
Tests the fit of the proportions in the obtained sample with the hypothesized proportions of the population.

7. Null hypothesis for Goodness of Fit
Specifies the proportion (or percentage) of the population in each category.
Rationale for null hypotheses:
No preference among categories.
No difference in one population from the proportions in another known population

8. Figure 16.1 Distribution of eye-color for a sample of n = 40

9. Data for the Goodness of Fit Test
In a sample of data, individuals in each category are counted.
Observed Frequencies in each category are measured.
Each individual is counted in one and only one category.

10. Expected frequencies in the Goodness of Fit Test
Goodness of Fit test compares the Observed Frequencies of the data with the assumptions of the null hypothesis.
Construct Expected Frequencies that are in perfect agreement with the null hypothesis.
Expected Frequency is the frequency value that is predicted from H0 and the sample size.
Ideal, hypothetical sample distribution

11. Chi-Square Statistics
Notation
χ2 is the lower-case Greek letter Chi
fo is the Observed Frequency
fe is the Expected Frequency
Chi-Square Statistic

12. Chi-Square distribution
Null hypothesis should be
Retained if the discrepancy between the Observed and Expected values is small
Rejected if the discrepancy between the Observed and Expected values is large
Chi-Square distribution includes values for all possible random samples when H0 is true
All chi-square values ≥ 0.
When H0 is true, Chi-square values will be small

13. Degrees of freedom and Chi-Square
Chi-square distribution is positively skewed
Chi-square is a family of distributions
Distributions determined by degrees of freedom
Slightly different shape for each value of df
Degrees of freedom for Goodness of Fit Test
df = C – 1
C is the number of categories

14. Figure 16.2 Chi-square distribution and the critical region

15. Figure 16.3 Chi-square distributions for different values of df

16. Critical region for a Chi-Square Test
Significance level is determined.
Critical value of chi-square is located in a table of critical values according to
Value for degrees of freedom (df)
Significance level chosen

17. Figure 16.4 Critical region for Example 16.1

18. Goodness of Fit and the Single-sample t Test
Both tests use data from one sample to test a hypothesis about a single population
Level of measurement determines test:
Numerical scores (interval / ratio scale) make it appropriate to compute a mean and use a t-test
Classification in nonnumeric categories (ordinal or nominal scale) make it appropriate to compute proportions or percentages and carry out a chi-square test

19. Learning Check The expected frequencies in a chi-square test ______.
are always whole numbers B
can contain fractions or decimal values C
can contain both positive and negative values D
can contain fractions and negative numbers

20. Learning Check - Answer
The expected frequencies in a chi-square test ______. A
are always whole numbers B
can contain fractions or decimal values C
can contain both positive and negative values D
can contain fractions and negative numbers

21. Learning Check
Decide if each of the following statements is True or False.
T/F
In a Chi-Square Test, the Observed Frequencies are always whole numbers.
A large value for chi-square will tend to retain the null hypothesis.

22. Answer
True
Observed frequencies are counts of individuals, so there are no fractional values.
False
Large values of chi-square indicate that the observed values were not very similar to what was predicted by the null hypothesis.

23. 16.3 Chi-Square Test for Independence
Chi-Square Statistic can test for the existence of a relationship between two variables.
Each individual classified on each variable
Counts are presented in the cells of a matrix
Research may be experimental or nonexperimental
Frequency data from a sample is used to evaluate the relationship of two variables in the population.

24. Null hypothesis for Test of Independence
Null hypothesis: two variables are independent
Two versions
Single population: No relationship between two variables in this population.
Two separate populations: No difference between distribution of variable in the two populations (defined by a nominal variable)
Variables are independent when there is no consistent predictable relationship between them.

25. Observed and expected frequencies
Frequencies in the sample are the Observed frequencies for the test.
Expected frequencies are based on the null hypothesis of same proportions in each category (population)
Proportions of each row total to the cells in each column

26. Computing expected frequencies
Frequencies computed by same method for each cell in the frequency distribution table
fc is frequency total for the column
fr is frequency total for the row

27. Chi-Square Statistic for Test of Independence
Same equation as the Chi-Square Test of Goodness of Fit
Chi-Square Statistic
Degrees of freedom df = (R-1)(C-1)
R is the number of rows
C is the number of columns

28. 16.4 Measuring effect size for Chi-Square
The Chi-square hypothesis test indicates that the difference did not occur by chance
Does not indicate the size of the effect
For a 2x2 matrix, the phi-coefficient Φ measures the strength of the relationship

29. Effect size in a larger matrix
For a larger matrix, a modification of the phi-coefficient is used: Cramer’s V
df* is the smaller of (R-1) or (C-1)

30. 16.5 Assumptions and restrictions for Chi-Square Tests
Independence of observations
Each observed frequency is generated by a different individual
Size of expected frequencies
Chi-square test should not be performed when the expected frequency of any cell is less than 5.

31. 16.6 Special applications for the Chi-Square Tests
Chi-square and Pearson correlation both evaluate relationships between two variables.
Type of data obtained determines which is the appropriate test to use.
Chi-square is sometimes used instead of t-tests or ANOVA, when counts rather than means of categories are being compared.
Chi-square can evaluate the significance.
Parametric tests measure strength and effect size with greater precision.

32. Learning Check
A basic assumption for a chi-square hypothesis test is ______. A
the population distribution(s) must be normal B
the scores must come from an interval or ratio scale C
the observations must be independent D
None of the other choices are assumptions for chi-square.

33. Learning Check - Answer
A basic assumption for a chi-square hypothesis test is ______. A
the population distribution(s) must be normal B
the scores must come from an interval or ratio scale C
the observations must be independent D
None of the other choices are assumptions for chi-square.

34. Learning Check
Decide if each of the following statements is True or False.
T/F
The value of df for a chi-square test does not depend on the sample size (n)
In the chi-square test for independence, a positive value for the chi-square statistic indicates a positive correlation between the two variables.

35. Any Questions?
Concepts?
Equations?