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Chapter 11

Chapter 11. Hypothesis Testing IV (Chi Square). 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. Chapter Outline.

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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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