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Chapter 11

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  1. Chapter 11 Hypothesis Testing IV (Chi Square)

  2. Chapter Outline • Introduction • Bivariate Tables • The Logic of Chi Square • The Computation of Chi Square • The Chi Square Test for Independence • The Chi Square Test: An Example

  3. Chapter Outline • An Additional Application of the Chi Square Test: The Goodness-of-Fit Test • The Limitations of the Chi Square Test • Interpreting Statistics: Family Values and Social Class

  4. In This Presentation • The basic logic of Chi Square. • The terminology used with bivariate tables. • The computation of Chi Square with an example problem. • The Five Step Model

  5. Basic Logic • Chi Square is a test of significance based on bivariate tables. • We are looking for significant differences between the actual cell frequencies in a table (fo) and those that would be expected by random chance (fe).

  6. Tables • Must have a title. • Cells are intersections of columns and rows. • Subtotals are called marginals. • N is reported at the intersection of row and column marginals.

  7. Tables • Columns are scores of the independent variable. • There will be as many columns as there are scores on the independent variable. • Rows are scores of the dependent variable. • There will be as many rows as there are scores on the dependent variable.

  8. Tables • There will be as many cells as there are scores on the two variables combined. • Each cell reports the number of times each combination of scores occurred.

  9. Tables

  10. Example of Computation • Problem 11.2 • Are the homicide rate and volume of gun sales related for a sample of 25 cities?

  11. Example of Computation • The bivariate table showing the relationship between homicide rate (columns) and gun sales (rows). This 2x2 table has 4 cells.

  12. Example of Computation • Use Formula 11.2 to find fe. • Multiply column and row marginals for each cell and divide by N. • For Problem 11.2 • (13*12)/25 = 156/25 = 6.24 • (13*13)/25 = 169/25 = 6.76 • (12*12)/25 = 144/25 = 5.76 • (12*13)/25 = 156/25 = 6.24

  13. Example of Computation • Expected frequencies:

  14. Example of Computation • A computational table helps organize the computations.

  15. Example of Computation • Subtract each fe from each fo. The total of this column must be zero.

  16. Example of Computation • Square each of these values

  17. Example of Computation • Divide each of the squared values by the fe for that cell. The sum of this column is chi square

  18. Step 1 Make Assumptions and Meet Test Requirements • Independent random samples • LOM is nominal • Note the minimal assumptions. In particular, note that no assumption is made about the shape of the sampling distribution. The chi square test is non-parametric.

  19. Step 2 State the Null Hypothesis • H0: The variables are independent • Another way to state the H0, more consistent with previous tests: • H0: fo = fe

  20. Step 2 State the Null Hypothesis • H1: The variables are dependent • Another way to state the H1: • H1: fo ≠ fe

  21. Step 3 Select the S. D. and Establish the C. R. • Sampling Distribution = χ2 • Alpha = .05 • df = (r-1)(c-1) = 1 • χ2 (critical) = 3.841

  22. Calculate the Test Statistic • χ2 (obtained) = 2.00

  23. Step 5 Make a Decision and Interpret the Results of the Test • χ2 (critical) = 3.841 • χ2 (obtained) = 2.00 • The test statistic is not in the Critical Region. Fail to reject the H0. • There is no significant relationship between homicide rate and gun sales.

  24. Interpreting Chi Square • The chi square test tells us only if the variables are independent or not. • It does not tell us the pattern or nature of the relationship. • To investigate the pattern, compute %s within each column and compare across the columns.

  25. Interpreting Chi Square • Cities low on homicide rate were high in gun sales and cities high in homicide rate were low in gun sales. • As homicide rates increase, gun sales decrease. This relationship is not significant but does have a clear pattern.

  26. The Limits of Chi Square • Like all tests of hypothesis, chi square is sensitive to sample size. • As N increases, obtained chi square increases. • With large samples, trivial relationships may be significant. • Remember: significance is not the same thing as importance.

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