Chi-Square Tests and Strategies When Population Distributions Are Not Normal <ul><li>Chapter 11 </li></ul>Copyright © 2011...
Chapter Outline <ul><li>Chi-Square Tests </li></ul><ul><li>The Chi-Square Statistic and the Chi-Square Test for Goodness o...
What If You Have Variables Whose Values Are Categories? <ul><li>t  Tests and the ANOVA require: </li></ul><ul><ul><li>the ...
Chi-Square Tests <ul><li>Are used when the variable of interest is a nominal variable </li></ul><ul><ul><li>The values of ...
Steps for Figuring the Chi-Square Statistic  <ul><li>Determine the actual observed frequencies in each category. </li></ul...
The Chi-Square Distribution <ul><li>Estimating the distribution of chi-square statistics that would arise by chance </li><...
The Chi-Square Table  (pg. 443) <ul><li>The cutoff for a chi-square to be extreme enough to reject the null hypothesis is ...
Summary: Hypothesis Testing Chi-Square Test for Goodness of Fit <ul><li>Restate the question as a research hypothesis and ...
Example of Hypothesis Testing Chi-Square Test for Goodness of Fit <ul><li>In the example from Chapter 11 of the text (pgs ...
<ul><li>Of the 1,386 characters whose gender was determined, 996 (72%) were male and 390 (28%) were female characters. </l...
Step 1: Restate the question as a H a  and H o <ul><ul><li>Population 1: characters on cereal boxes like those in the stud...
Step 2: Determine the characteristics of the comparison distribution. <ul><ul><li>Chi-squared distribution </li></ul></ul>...
Step 3: Determine the cutoff on the comparison distribution at which the null hypothesis should be rejected. <ul><ul><li>L...
Step 4: Determine your sample’s score on the comparison distribution. <ul><ul><li>Determine the actual observed frequencie...
Step 4: Determine your sample’s score on the comparison distribution. <ul><ul><li>Determine the actual observed frequencie...
Step 4: Determine your sample’s score on the comparison distribution. <ul><ul><li>Determine the expected frequencies in ea...
Step 4: Determine your sample’s score on the comparison distribution. <ul><ul><li>In each category, take observed minus ex...
Step 4: Determine your sample’s score on the comparison distribution. <ul><ul><li>Square each of these differences. </li><...
Step 4: Determine your sample’s score on the comparison distribution. <ul><ul><li>Divide each squared difference by the ex...
Step 4: Determine your sample’s score on the comparison distribution. <ul><ul><li>Add up these results for all categories....
Example of Hypothesis Testing Chi-Square Test for Goodness of Fit: Step 5 <ul><li>Decide whether to reject the null hypoth...
Second Example of Hypothesis Testing  Chi-Square Test for Goodness of Fit <ul><li>Chapter 11 Study Guide Problem 1 (pg. 17...
Step 1: Restate the question as a H a  and H o <ul><ul><li>Population 1 : the population of students who use each method o...
Step 2: Determine the characteristics of the comparison distribution. <ul><ul><li>Chi-squared distribution </li></ul></ul>...
Step 3: Determine the cutoff on the comparison distribution at which the null hypothesis should be rejected. <ul><ul><li>L...
Step 4: Determine your sample’s score on the comparison distribution. <ul><ul><li>Determine the actual observed frequencie...
Step 4: Determine your sample’s score on the comparison distribution. <ul><ul><li>Determine the expected frequencies in ea...
Step 4: Determine your sample’s score on the comparison distribution. <ul><ul><li>In each category, take observed minus ex...
Step 4: Determine your sample’s score on the comparison distribution. <ul><ul><li>Square each of these differences. </li><...
Step 4: Determine your sample’s score on the comparison distribution. <ul><ul><li>Divide each squared difference by the ex...
Step 4: Determine your sample’s score on the comparison distribution. <ul><ul><li>Add up these results for all categories....
Example of Hypothesis Testing Chi-Square Test for Goodness of Fit: Step 5 <ul><li>Decide whether to reject the null hypoth...
The Chi-Square Test for Independence <ul><li>Used when there are two nominal variables, each with several categories </li>...
Independence <ul><li>No relationship between the variables in the contingency table </li></ul><ul><li>It is important to d...
Determining Expected Frequencies
Figuring the Chi-Square <ul><li>Figure the weighted squared difference for each cell and add these up </li></ul>Copyright ...
Degrees of Freedom <ul><li>With a chi-square test for independence, the degrees of freedom are the number of categories fr...
Steps of Hypothesis Testing <ul><li>Restate the question as a research hypothesis and a null hypothesis about the populati...
Example of Steps of Hypothesis Testing: Step 1 <ul><li>Restate the question as a research hypothesis and a null hypothesis...
Step 2: Determine the characteristics of the comparison distribution. <ul><ul><li>Chi-square distribution </li></ul></ul><...
Step 3: Determine the cutoff sample score on the comparison distribution at which the null hypothesis should be rejected. ...
Step 4: Determine your sample’s score on the comparison distribution. <ul><ul><li>Determine the actual observed frequencie...
Step 4: Determine your sample’s score on the comparison distribution. <ul><ul><li>EXPECTED FREQUENCIES </li></ul></ul><ul>...
<ul><li>Determine your sample’s score on the comparison distribution. </li></ul><ul><ul><li>Figure the X 2 </li></ul></ul>...
+ + +   Gender TOTAL Age  Male Female   O E O E <ul><ul><li>Child </li></ul></ul>28 39.9 30 18.0 58 (26.1%) <ul><ul><li>Ad...
Step 5:  Decide whether to accept or reject the null hypothesis. <ul><ul><li>Your sample is  X 2  =  15.62 ; this is great...
Second Example of Hypothesis Testing  Chi-Square Test for Independence <ul><li>Chapter 11 Study Guide Problem 2 (pg. 170) ...
Step 1: Restate the question as a research hypothesis and a null hypothesis about the population. <ul><ul><li>Population 1...
Step 2: Determine the characteristics of the comparison distribution. <ul><ul><li>Chi-square distribution </li></ul></ul><...
Step 3: Determine the cutoff sample score on the comparison distribution at which the null hypothesis should be rejected. ...
Step 4: Determine your sample’s score on the comparison distribution. <ul><ul><li>Determine the actual observed frequencie...
Step 4: Determine your sample’s score on the comparison distribution. <ul><ul><li>EXPECTED FREQUENCIES </li></ul></ul><ul>...
<ul><li>Determine your sample’s score on the comparison distribution. </li></ul><ul><ul><li>Figure the X 2 </li></ul></ul>...
Suspended  Aggressive  Manipulative  Passive  Assertive  TOTAL  Yes 7  (4) 1  (1) 1  (1) 1  (4) 10 (50%) No  1  (4) 1  (1)...
Step 5:  Decide whether to accept or reject the null hypothesis. <ul><ul><li>Your sample is  X 2  =  9 ; this is greater t...
Assumptions for the Chi-Square Tests <ul><li>Each score must not have any special relationship to any other score. </li></...
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Aron chpt 11 ed (2)

  1. 1. Chi-Square Tests and Strategies When Population Distributions Are Not Normal <ul><li>Chapter 11 </li></ul>Copyright © 2011 by Pearson Education, Inc. All rights reserved
  2. 2. Chapter Outline <ul><li>Chi-Square Tests </li></ul><ul><li>The Chi-Square Statistic and the Chi-Square Test for Goodness of Fit </li></ul><ul><li>The Chi-Square Test for Independence </li></ul><ul><li>Assumptions of the Chi-Square Tests </li></ul><ul><li>Effect Size and Power for the Chi-Square Tests for Independence </li></ul><ul><li>Strategies for Hypothesis Testing When Population Distributions Are Not Normal </li></ul><ul><li>Data Transformations </li></ul><ul><li>Rank-Order Tests </li></ul><ul><li>Comparison of Methods </li></ul><ul><li>Chi-Square Tests, Data Transformation, and Rank-Order Tests in Research Articles </li></ul>Copyright © 2011 by Pearson Education, Inc. All rights reserved
  3. 3. What If You Have Variables Whose Values Are Categories? <ul><li>t Tests and the ANOVA require: </li></ul><ul><ul><li>the measured variable to have scores that are quantitative </li></ul></ul><ul><ul><ul><li>e.g., ratings on a scale of stress that range from 0–10, numerical scores on a test of intelligence, scores on a measure of gastrointestinal symptoms </li></ul></ul></ul><ul><ul><li>the populations to follow a normal curve. </li></ul></ul><ul><li>Categorical variables require an alternative hypothesis-testing procedure. </li></ul>Copyright © 2011 by Pearson Education, Inc. All rights reserved
  4. 4. Chi-Square Tests <ul><li>Are used when the variable of interest is a nominal variable </li></ul><ul><ul><li>The values of a nominal variable are categories. </li></ul></ul><ul><ul><li>The scores of a nominal variable represent frequencies. </li></ul></ul><ul><li>Chi-square tests examine how well the observed breakdown of people or observations over categories fits an expected breakdown. </li></ul><ul><ul><li>Chi-square test of goodness of fit involves levels of a single nominal variable. </li></ul></ul><ul><ul><li>Chi-square test for independence is used when there are two nominal variables each with several categories. </li></ul></ul>Copyright © 2011 by Pearson Education, Inc. All rights reserved
  5. 5. Steps for Figuring the Chi-Square Statistic <ul><li>Determine the actual observed frequencies in each category. </li></ul><ul><li>Determine the expected frequency in each category. </li></ul><ul><li>In each category, take the observed minus expected frequencies. </li></ul><ul><li>Square each of these differences. </li></ul><ul><li>Divide each squared difference by the expected frequency for its category. </li></ul><ul><li>Add up the these results for all the categories. </li></ul>
  6. 6. The Chi-Square Distribution <ul><li>Estimating the distribution of chi-square statistics that would arise by chance </li></ul><ul><li>The exact shape of the chi-square distribution depends on degrees of freedom, but they are all skewed to the right because the chi-square statistic cannot be less than 0 but can have very high values. </li></ul><ul><li>The degrees of freedom for a chi-square test are the number of categories that are free to vary, given the total. </li></ul><ul><ul><li>For example, if there are two categories, there is one degree of freedom. </li></ul></ul><ul><ul><ul><li>df = N categories – 1 </li></ul></ul></ul>Copyright © 2011 by Pearson Education, Inc. All rights reserved
  7. 7. The Chi-Square Table (pg. 443) <ul><li>The cutoff for a chi-square to be extreme enough to reject the null hypothesis is determined using a chi-square table. </li></ul><ul><ul><li>To use this table, you need to determine the degrees of freedom and the significance level that you will use for your study. </li></ul></ul>
  8. 8. Summary: Hypothesis Testing Chi-Square Test for Goodness of Fit <ul><li>Restate the question as a research hypothesis and a null hypothesis about the population. </li></ul><ul><ul><li>The research hypothesis is that the observations over categories in the two populations are different. </li></ul></ul><ul><ul><li>The null hypothesis is that the observations over categories in the two populations are the same. </li></ul></ul><ul><li>Determine the characteristics of the comparison distribution. </li></ul><ul><ul><li>chi-squared distribution; df = number of categories – 1 </li></ul></ul><ul><li>Determine the cutoff on the comparison distribution at which the null hypothesis should be rejected. </li></ul><ul><li>Determine your sample’s score on the comparison distribution. </li></ul><ul><ul><li>Determine the actual observed frequencies in each category. </li></ul></ul><ul><ul><li>Determine the expected frequencies in each category. </li></ul></ul><ul><ul><li>In each category, take observed minus expected frequencies. </li></ul></ul><ul><ul><li>Square each of these differences. </li></ul></ul><ul><ul><li>Divide each squared difference by the expected frequency for its category. </li></ul></ul><ul><ul><li>Add up these results for all categories. </li></ul></ul><ul><li>Decide whether to reject the null hypothesis. </li></ul><ul><ul><li>Compare your sample’s score to the cutoff score. </li></ul></ul>
  9. 9. Example of Hypothesis Testing Chi-Square Test for Goodness of Fit <ul><li>In the example from Chapter 11 of the text (pgs 366-372), the researchers looked at the gender of characters on cereal boxes. </li></ul><ul><li>“ In order to test their hypothesis that male characters would appear more often than female characters on cereal boxes, researchers coded the gender of the characters on every cereal box in a large grocery superstore” </li></ul>Copyright © 2011 by Pearson Education, Inc. All rights reserved
  10. 10. <ul><li>Of the 1,386 characters whose gender was determined, 996 (72%) were male and 390 (28%) were female characters. </li></ul>Example of Hypothesis Testing Chi-Square Test for Goodness of Fit Copyright © 2011 by Pearson Education, Inc. All rights reserved
  11. 11. Step 1: Restate the question as a H a and H o <ul><ul><li>Population 1: characters on cereal boxes like those in the study </li></ul></ul><ul><ul><li>Population 2: characters on cereal boxes who are equally likely to be male and female </li></ul></ul><ul><ul><li>H a = Male characters will appear more often than females characters on cereal boxes. </li></ul></ul><ul><ul><li>(the distribution of observations over categories in the 2 populations is different) </li></ul></ul><ul><ul><li>H o = There will be no difference is the appearance of male and female characters on cereal boxes . </li></ul></ul><ul><ul><li>(the distribution of observations over categories in the 2 populations is not different) </li></ul></ul>
  12. 12. Step 2: Determine the characteristics of the comparison distribution. <ul><ul><li>Chi-squared distribution </li></ul></ul><ul><ul><li>Degrees of Freedom = Number of Categories –1 </li></ul></ul><ul><ul><li>2 categories (male & female characters) -1 = 1 </li></ul></ul>Copyright © 2011 by Pearson Education, Inc. All rights reserved
  13. 13. Step 3: Determine the cutoff on the comparison distribution at which the null hypothesis should be rejected. <ul><ul><li>Look at the cutoff on the chi-square table for your significance level and the study’s degrees of freedom. </li></ul></ul><ul><ul><li>Using a significance level of .05 and 1 degree of freedom the cutoff from the chi-square table is 3.841. </li></ul></ul>Copyright © 2011 by Pearson Education, Inc. All rights reserved
  14. 14. Step 4: Determine your sample’s score on the comparison distribution. <ul><ul><li>Determine the actual observed frequencies in each category. </li></ul></ul><ul><ul><ul><li>males = 996, females = 390 </li></ul></ul></ul><ul><ul><li>Determine the expected frequencies in each category. </li></ul></ul><ul><ul><ul><li>males = 693, females = 693 </li></ul></ul></ul><ul><ul><li>In each category, take observed minus expected frequencies. </li></ul></ul><ul><ul><ul><li>for males: O – E = 996 – 693= 303 </li></ul></ul></ul><ul><ul><ul><li>for females: O – E = 390 – 693= -303 </li></ul></ul></ul><ul><ul><li>Square each of these differences. </li></ul></ul><ul><ul><ul><li>for males: (O – E) 2 = (303) 2 = 91,809 </li></ul></ul></ul><ul><ul><ul><li>for females: (O – E) 2 = (-303) 2 = 91,809 </li></ul></ul></ul><ul><ul><li>Divide each squared difference by the expected frequency for its category. </li></ul></ul><ul><ul><li>for males: (O-E)2 /E = 132.48 </li></ul></ul><ul><ul><li>for females: (O-E)2 / E = 132.48 </li></ul></ul><ul><ul><li>Add up these results for all categories. </li></ul></ul><ul><ul><li>X 2 = 132.48 + 132.48 = 264.9 </li></ul></ul>Copyright © 2011 by Pearson Education, Inc. All rights reserved
  15. 15. Step 4: Determine your sample’s score on the comparison distribution. <ul><ul><li>Determine the actual observed frequencies in each category. </li></ul></ul><ul><ul><ul><li>males = 996, females = 390 </li></ul></ul></ul>  O E (O-E) (O-E) 2 Male 996 693 303 91,809 91809/693 132.48 Female 390 693 -303 91,809 91809/693 132.48 264.96
  16. 16. Step 4: Determine your sample’s score on the comparison distribution. <ul><ul><li>Determine the expected frequencies in each category. </li></ul></ul><ul><ul><ul><li>males = 693, females = 693 </li></ul></ul></ul>  O E (O-E) (O-E) 2 Male 996 693 303 91,809 91809/693 132.48 Female 390 693 -303 91,809 91809/693 132.48 264.96
  17. 17. Step 4: Determine your sample’s score on the comparison distribution. <ul><ul><li>In each category, take observed minus expected frequencies. </li></ul></ul>  O E (O-E) (O-E) 2 Male 996 693 303 91,809 91809/693 132.48 Female 390 693 -303 91,809 91809/693 132.48 264.96
  18. 18. Step 4: Determine your sample’s score on the comparison distribution. <ul><ul><li>Square each of these differences. </li></ul></ul>  O E (O-E) (O-E) 2 Male 996 693 303 91,809 91809/693 132.48 Female 390 693 -303 91,809 91809/693 132.48 264.96
  19. 19. Step 4: Determine your sample’s score on the comparison distribution. <ul><ul><li>Divide each squared difference by the expected frequency for its category. </li></ul></ul>  O E (O-E) (O-E) 2 Male 996 693 303 91,809 91809/693 132.48 Female 390 693 -303 91,809 91809/693 132.48 264.96
  20. 20. Step 4: Determine your sample’s score on the comparison distribution. <ul><ul><li>Add up these results for all categories. </li></ul></ul>  O E (O-E) (O-E) 2 Male 996 693 303 91,809 91809/693 132.48 Female 390 693 -303 91,809 91809/693 132.48 264.96
  21. 21. Example of Hypothesis Testing Chi-Square Test for Goodness of Fit: Step 5 <ul><li>Decide whether to reject the null hypothesis. </li></ul><ul><ul><li>The chi-square of the sample—264.96—is greater than the cutoff to reject the null hypothesis, which is 3.841. </li></ul></ul><ul><ul><li>The researchers can reject the null hypothesis. </li></ul></ul>Copyright © 2011 by Pearson Education, Inc. All rights reserved
  22. 22. Second Example of Hypothesis Testing Chi-Square Test for Goodness of Fit <ul><li>Chapter 11 Study Guide Problem 1 (pg. 170) </li></ul><ul><li>The table that follows includes the primary method of conflict resolution used by 20 students. </li></ul><ul><li>(a) Following the five steps of hypothesis testing, conduct the appropriate X 2 test to determine whether the observed frequencies are significantly different from the frequencies expected at the .05 level of significance. (b) Explain your results. </li></ul>Method Aggressive Manipulative Passive Assertive N of students 8 2 2 8
  23. 23. Step 1: Restate the question as a H a and H o <ul><ul><li>Population 1 : the population of students who use each method of conflict resolution like those observed. </li></ul></ul><ul><ul><li>Population 2 : the population of students who use each method of conflict resolution equally. </li></ul></ul><ul><ul><li>H a = Students will use the methods of conflict resolution differently. </li></ul></ul><ul><ul><li>(the distribution of observations over categories in the 2 populations is different) </li></ul></ul><ul><ul><li>H o = Students will not use the methods of conflict resolution differently . </li></ul></ul><ul><ul><li>(the distribution of observations over categories in the 2 populations is not different) </li></ul></ul>
  24. 24. Step 2: Determine the characteristics of the comparison distribution. <ul><ul><li>Chi-squared distribution </li></ul></ul><ul><ul><li>Degrees of Freedom = Number of Categories –1 </li></ul></ul><ul><ul><li>4 categories (methods of conflict resolution) -1 = 3 </li></ul></ul>Copyright © 2011 by Pearson Education, Inc. All rights reserved
  25. 25. Step 3: Determine the cutoff on the comparison distribution at which the null hypothesis should be rejected. <ul><ul><li>Look at the cutoff on the chi-square table for your significance level and the study’s degrees of freedom. </li></ul></ul><ul><ul><li>Using a significance level of .05 and 3 degree of freedom the cutoff from the chi-square table is 7.815. </li></ul></ul>Copyright © 2011 by Pearson Education, Inc. All rights reserved
  26. 26. Step 4: Determine your sample’s score on the comparison distribution. <ul><ul><li>Determine the actual observed frequencies in each category. </li></ul></ul><ul><ul><ul><li>Aggressive = 8 </li></ul></ul></ul><ul><ul><ul><li>Manipulative = 2 </li></ul></ul></ul><ul><ul><ul><li>Passive = 2 </li></ul></ul></ul><ul><ul><ul><li>Assertive = 8 </li></ul></ul></ul>  O E (O-E) (O-E) 2     <ul><ul><li>Aggressive </li></ul></ul>8 5 3 9 9/5 1.8 <ul><ul><li>Manipulative </li></ul></ul>2 5 -3 9 9/6 1.8 <ul><ul><li>Passive </li></ul></ul>2 5 3 9 9/7 1.8 <ul><ul><li>Assertive </li></ul></ul>8 5 -3 9 9/8 1.8             7.2
  27. 27. Step 4: Determine your sample’s score on the comparison distribution. <ul><ul><li>Determine the expected frequencies in each category. </li></ul></ul><ul><ul><ul><li>Aggressive = 5 </li></ul></ul></ul><ul><ul><ul><li>Manipulative = 5 </li></ul></ul></ul><ul><ul><ul><li>Passive = 5 </li></ul></ul></ul><ul><ul><ul><li>Assertive = 5 </li></ul></ul></ul>  O E (O-E) (O-E) 2     <ul><ul><li>Aggressive </li></ul></ul>8 5 3 9 9/5 1.8 <ul><ul><li>Manipulative </li></ul></ul>2 5 -3 9 9/6 1.8 <ul><ul><li>Passive </li></ul></ul>2 5 3 9 9/7 1.8 <ul><ul><li>Assertive </li></ul></ul>8 5 -3 9 9/8 1.8             7.2
  28. 28. Step 4: Determine your sample’s score on the comparison distribution. <ul><ul><li>In each category, take observed minus expected frequencies. </li></ul></ul>  O E (O-E) (O-E) 2     <ul><ul><li>Aggressive </li></ul></ul>8 5 3 9 9/5 1.8 <ul><ul><li>Manipulative </li></ul></ul>2 5 -3 9 9/6 1.8 <ul><ul><li>Passive </li></ul></ul>2 5 3 9 9/7 1.8 <ul><ul><li>Assertive </li></ul></ul>8 5 -3 9 9/8 1.8             7.2
  29. 29. Step 4: Determine your sample’s score on the comparison distribution. <ul><ul><li>Square each of these differences. </li></ul></ul>  O E (O-E) (O-E) 2     <ul><ul><li>Aggressive </li></ul></ul>8 5 3 9 9/5 1.8 <ul><ul><li>Manipulative </li></ul></ul>2 5 -3 9 9/6 1.8 <ul><ul><li>Passive </li></ul></ul>2 5 3 9 9/7 1.8 <ul><ul><li>Assertive </li></ul></ul>8 5 -3 9 9/8 1.8             7.2
  30. 30. Step 4: Determine your sample’s score on the comparison distribution. <ul><ul><li>Divide each squared difference by the expected frequency for its category. </li></ul></ul>  O E (O-E) (O-E) 2     <ul><ul><li>Aggressive </li></ul></ul>8 5 3 9 9/5 1.8 <ul><ul><li>Manipulative </li></ul></ul>2 5 -3 9 9/6 1.8 <ul><ul><li>Passive </li></ul></ul>2 5 3 9 9/7 1.8 <ul><ul><li>Assertive </li></ul></ul>8 5 -3 9 9/8 1.8             7.2
  31. 31. Step 4: Determine your sample’s score on the comparison distribution. <ul><ul><li>Add up these results for all categories. </li></ul></ul>  O E (O-E) (O-E) 2     <ul><ul><li>Aggressive </li></ul></ul>8 5 3 9 9/5 1.8 <ul><ul><li>Manipulative </li></ul></ul>2 5 -3 9 9/6 1.8 <ul><ul><li>Passive </li></ul></ul>2 5 3 9 9/7 1.8 <ul><ul><li>Assertive </li></ul></ul>8 5 -3 9 9/8 1.8             7.2
  32. 32. Example of Hypothesis Testing Chi-Square Test for Goodness of Fit: Step 5 <ul><li>Decide whether to reject the null hypothesis. </li></ul><ul><ul><li>The chi-square of the sample—7.2—is not more extreme than the cutoff to reject the null hypothesis, which is 7.815. </li></ul></ul><ul><ul><li>The researchers cannot reject the null hypothesis. </li></ul></ul>Copyright © 2011 by Pearson Education, Inc. All rights reserved
  33. 33. The Chi-Square Test for Independence <ul><li>Used when there are two nominal variables, each with several categories </li></ul><ul><li>Contingency Table </li></ul><ul><ul><li>table in which the distributions of two nominal variables are set up so that you have the frequencies of their contributions as well as the totals </li></ul></ul>Copyright © 2011 by Pearson Education, Inc. All rights reserved   Gender TOTAL Age  Male Female   <ul><ul><li>Child </li></ul></ul>28 30 58 (26.1%) <ul><ul><li>Adult </li></ul></ul>125 39 164 (73.9%) <ul><ul><li>TOTAL </li></ul></ul>153 69 222 (100.0%)
  34. 34. Independence <ul><li>No relationship between the variables in the contingency table </li></ul><ul><li>It is important to determine whether the lack of independence in the sample is large enough to reject the null hypothesis of independence of the population. </li></ul>  Gender TOTAL Age  Male Female   <ul><ul><li>Child </li></ul></ul>28 (18.3%) 30 (43.5%) 58 (26.1%) <ul><ul><li>Adult </li></ul></ul>125 (81.7%) 39 (56.5%) 128 (73.9%) <ul><ul><li>TOTAL </li></ul></ul>153 69 222 100.0%)
  35. 35. Determining Expected Frequencies
  36. 36. Figuring the Chi-Square <ul><li>Figure the weighted squared difference for each cell and add these up </li></ul>Copyright © 2011 by Pearson Education, Inc. All rights reserved
  37. 37. Degrees of Freedom <ul><li>With a chi-square test for independence, the degrees of freedom are the number of categories free to vary once the totals are known. </li></ul><ul><ul><li>The number of categories is the number of cells. </li></ul></ul><ul><ul><li>The totals include the row and column totals. </li></ul></ul><ul><ul><li>df = (N Columns – 1)(N Rows – 1) </li></ul></ul>Copyright © 2011 by Pearson Education, Inc. All rights reserved
  38. 38. Steps of Hypothesis Testing <ul><li>Restate the question as a research hypothesis and a null hypothesis about the population. </li></ul><ul><li>Determine the characteristics of the comparison distribution. </li></ul><ul><ul><li>chi-square distribution </li></ul></ul><ul><ul><li>degrees of freedom = (N Columns – 1)(N Rows – 1) </li></ul></ul><ul><li>Determine the cutoff sample score on the comparison distribution at which the null hypothesis should be rejected. </li></ul><ul><ul><li>Decide the significance level you will use for your study. </li></ul></ul><ul><ul><li>Use the degrees of freedom you calculated and the significance level you will use for this study to find the cutoff score on a chi-square table. </li></ul></ul><ul><li>Determine your sample’s score on the comparison distribution. </li></ul><ul><ul><li>Determine the actual observed frequencies in each cell. </li></ul></ul><ul><ul><li>Determine the expected frequencies in each cell. </li></ul></ul><ul><ul><ul><li>Find each row’s percentage of the total. </li></ul></ul></ul><ul><ul><ul><li>For each row, multiply its rows percentage by its column’s total. </li></ul></ul></ul><ul><ul><ul><li>In each cell, take the observed minus the expected frequencies. </li></ul></ul></ul><ul><ul><ul><li>Square each of these differences. </li></ul></ul></ul><ul><ul><ul><li>Divide each squared difference by the expected frequency for its cell. </li></ul></ul></ul><ul><ul><ul><li>Add up the results for all the cells. </li></ul></ul></ul><ul><li>Decide whether to accept or reject the null hypothesis. </li></ul>
  39. 39. Example of Steps of Hypothesis Testing: Step 1 <ul><li>Restate the question as a research hypothesis and a null hypothesis about the population. </li></ul><ul><ul><li>Population 1: characters on cereal boxes like those in the study </li></ul></ul><ul><ul><li>Population 2: characters on cereal boxes for which the ages distribution of the characters is independent of the gender of the characters </li></ul></ul><ul><ul><li>H a : The proportion of characters that are children and adults are different for male and female characters. </li></ul></ul><ul><ul><li>H o : The proportion of characters that are children and adults are not different for male and female characters. </li></ul></ul>Copyright © 2011 by Pearson Education, Inc. All rights reserved
  40. 40. Step 2: Determine the characteristics of the comparison distribution. <ul><ul><li>Chi-square distribution </li></ul></ul><ul><ul><li>Degrees of freedom = (N Columns – 1)(N Rows – 1) </li></ul></ul><ul><ul><li>df = (N Gender – 1)(N Age – 1) </li></ul></ul><ul><ul><li>df = (2-1)(2-1) </li></ul></ul><ul><ul><li>df = 1 </li></ul></ul>  Gender TOTAL Age  Male Female   <ul><ul><li>Child </li></ul></ul>28 30 58 (26.1%) <ul><ul><li>Adult </li></ul></ul>125 39 164 (73.9%) <ul><ul><li>TOTAL </li></ul></ul>153 69 222 (100.0%)
  41. 41. Step 3: Determine the cutoff sample score on the comparison distribution at which the null hypothesis should be rejected. <ul><ul><li>Decide the significance level you will use for your study. </li></ul></ul><ul><ul><li>For 1 degree of freedom and a .05 significance level, using the chi-square table your cutoff is 3.841 </li></ul></ul>
  42. 42. Step 4: Determine your sample’s score on the comparison distribution. <ul><ul><li>Determine the actual observed frequencies in each cell. </li></ul></ul><ul><ul><ul><li>male/child cell O = 28 </li></ul></ul></ul><ul><ul><ul><li>male/adult cell O = 125 </li></ul></ul></ul><ul><ul><ul><li>female/child cell O = 30 </li></ul></ul></ul><ul><ul><ul><li>female/adult cell O= 39 </li></ul></ul></ul><ul><ul><li>Find each row’s percentage of the total. </li></ul></ul><ul><ul><ul><li>total for the child row = 58 </li></ul></ul></ul><ul><ul><ul><li>total for the adult row = 164 </li></ul></ul></ul><ul><ul><ul><li>total = 222 </li></ul></ul></ul><ul><ul><ul><li>child row percentage = 58 / 222 = 26.1% </li></ul></ul></ul><ul><ul><ul><li>adult row percentage = 164 / 222 = 73.9% </li></ul></ul></ul>  Gender TOTAL Age  Male Female   <ul><ul><li>Child </li></ul></ul>28 30 58 (26.1%) <ul><ul><li>Adult </li></ul></ul>125 39 164 (73.9%) <ul><ul><li>TOTAL </li></ul></ul>153 69 222
  43. 43. Step 4: Determine your sample’s score on the comparison distribution. <ul><ul><li>EXPECTED FREQUENCIES </li></ul></ul><ul><ul><li>For each cell, multiply its row’s percentage by the column’s total. </li></ul></ul>  Gender TOTAL Age  Male Female   <ul><ul><li>Child </li></ul></ul>28 (.261)(153)= 39.9 30 (.261)(69)= 18.0 58 (26.1%) <ul><ul><li>Adult </li></ul></ul>125 (.739)(153)= 113.1 39 (.739)(69)= 51.0 164 (73.9%) <ul><ul><li>TOTAL </li></ul></ul>153 69 222
  44. 44. <ul><li>Determine your sample’s score on the comparison distribution. </li></ul><ul><ul><li>Figure the X 2 </li></ul></ul>Step 4: Determine your sample’s score on the comparison distribution.   Gender TOTAL Age  Male Female   O E O E <ul><ul><li>Child </li></ul></ul>28 39.9 30 18.0 58 (26.1%) <ul><ul><li>Adult </li></ul></ul>125 113.1 39 51.0 164 (73.9%) <ul><ul><li>TOTAL </li></ul></ul>153 69 222
  45. 45. + + +   Gender TOTAL Age  Male Female   O E O E <ul><ul><li>Child </li></ul></ul>28 39.9 30 18.0 58 (26.1%) <ul><ul><li>Adult </li></ul></ul>125 113.1 39 51.0 164 (73.9%) <ul><ul><li>TOTAL </li></ul></ul>153 69 222
  46. 46. Step 5: Decide whether to accept or reject the null hypothesis. <ul><ul><li>Your sample is X 2 = 15.62 ; this is greater than 3.841 , the score needed to reject the null hypothesis. </li></ul></ul><ul><ul><li>You can reject the null hypothesis. </li></ul></ul><ul><ul><li>H a : The proportion of characters that are children and adults are different for male and female characters. </li></ul></ul>
  47. 47. Second Example of Hypothesis Testing Chi-Square Test for Independence <ul><li>Chapter 11 Study Guide Problem 2 (pg. 170) </li></ul><ul><li>The behavioral scientists categorized the students based on the primary method of conflict resolution used and whether the student had been suspended from school for misbehavior. </li></ul><ul><li>(a) Following the five steps of hypothesis testing, conduct the appropriate X 2 test to determine whether the observed frequencies are significantly different from the frequencies expected at the .05 level of significance. (b) Explain your results. </li></ul>Suspended Aggressive Manipulative Passive Assertive TOTAL Yes 7 1 1 1 10 No 1 1 1 7 10 TOTAL 8 2 2 8 20
  48. 48. Step 1: Restate the question as a research hypothesis and a null hypothesis about the population. <ul><ul><li>Population 1: Students for whom the primary method of conflict resolution is associated with being or not being suspended from school for misbehavior. </li></ul></ul><ul><ul><li>Population 2: Students for whom the primary method of conflict resolution is independent of being or not being suspended from school for misbehavior. </li></ul></ul><ul><ul><li>H a : The primary method of conflict resolution used by students who have been suspended from school are different from the methods of students who have not been suspended. </li></ul></ul><ul><ul><li>H o : The primary method of conflict resolution used by students who have been suspended from school are no different from the methods of students who have not been suspended. </li></ul></ul>
  49. 49. Step 2: Determine the characteristics of the comparison distribution. <ul><ul><li>Chi-square distribution </li></ul></ul><ul><ul><li>Degrees of freedom = (N Columns – 1)(N Rows – 1) </li></ul></ul><ul><ul><li>df = (N Method – 1)(N Suspended – 1) </li></ul></ul><ul><ul><li>df = (4-1)(2-1) </li></ul></ul><ul><ul><li>df = 3 </li></ul></ul>Suspended Aggressive Manipulative Passive Assertive TOTAL Yes 7 1 1 1 10 No 1 1 1 7 10 TOTAL 8 2 2 8 20
  50. 50. Step 3: Determine the cutoff sample score on the comparison distribution at which the null hypothesis should be rejected. <ul><ul><li>Decide the significance level you will use for your study. </li></ul></ul><ul><ul><li>For 1 degree of freedom and a .05 significance level, using the chi-square table your cutoff is 7.815 </li></ul></ul>
  51. 51. Step 4: Determine your sample’s score on the comparison distribution. <ul><ul><li>Determine the actual observed frequencies in each cell. </li></ul></ul><ul><ul><li>Find each row’s percentage of the total. </li></ul></ul>Suspended Aggressive Manipulative Passive Assertive TOTAL Yes 7 1 1 1 10 (50%) No 1 1 1 7 10 (50%) TOTAL 8 2 2 8 20
  52. 52. Step 4: Determine your sample’s score on the comparison distribution. <ul><ul><li>EXPECTED FREQUENCIES </li></ul></ul><ul><ul><li>For each cell, multiply its row’s percentage by the column’s total. </li></ul></ul>Suspended Aggressive Manipulative Passive Assertive TOTAL Yes 7 (4) 1 (1) 1 (1) 1 (4) 10 (50%) No 1 (4) 1 (1) 1 (1) 7 (4) 10 (50%) TOTAL 8 2 2 8 20
  53. 53. <ul><li>Determine your sample’s score on the comparison distribution. </li></ul><ul><ul><li>Figure the X 2 </li></ul></ul>Step 4: Determine your sample’s score on the comparison distribution. Suspended Aggressive Manipulative Passive Assertive TOTAL Yes 7 (4) 1 (1) 1 (1) 1 (4) 10 (50%) No 1 (4) 1 (1) 1 (1) 7 (4) 10 (50%) TOTAL 8 2 2 8 20
  54. 54. Suspended Aggressive Manipulative Passive Assertive TOTAL Yes 7 (4) 1 (1) 1 (1) 1 (4) 10 (50%) No 1 (4) 1 (1) 1 (1) 7 (4) 10 (50%) TOTAL 8 2 2 8 20
  55. 55. Step 5: Decide whether to accept or reject the null hypothesis. <ul><ul><li>Your sample is X 2 = 9 ; this is greater than 7.815 , the score needed to reject the null hypothesis. </li></ul></ul><ul><ul><li>You can reject the null hypothesis. </li></ul></ul><ul><ul><li>H a : The primary method of conflict resolution used by students who have been suspended from school are different from the methods of students who have not been suspended. </li></ul></ul>
  56. 56. Assumptions for the Chi-Square Tests <ul><li>Each score must not have any special relationship to any other score. </li></ul><ul><ul><li>You cannot use the chi-square tests if the scores are based on the same people being tested more than once. </li></ul></ul>Copyright © 2011 by Pearson Education, Inc. All rights reserved
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