Chapter 16 The Chi-Square Statistic

1. Chapter 16The Chi-Square Statistic PowerPoint Lecture SlidesEssentials of Statistics for the Behavioral SciencesSeventh Editionby Frederick J. Gravetter and Larry B. Wallnau

2. Chapter 16 Learning Outcomes

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 categoryare 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 • χ2is the lower-case Greek letter Chi • fois the Observed Frequency • feis 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 • Cis 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 ______.

20. Learning Check - Answer • The expected frequencies in a chi-square test ______.

21. Learning Check • Decide if each of the following statements is True or False.

22. Answer

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-coefficientΦ 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 adifferent 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 ______.

33. Learning Check - Answer • A basic assumption for a chi-square hypothesis test is ______.

34. Learning Check • Decide if each of the following statements is True or False.