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  1. 1. Difference of the Means Andrew Martin PS 372 University of Kentucky
  2. 2. ANOVA <ul><li>For quantitative (or interval- and ratio-level ) data the analysis of variance is appropriate. </li></ul><ul><li>Analysis of variance aka ANOVA. </li></ul><ul><li>The independent variable, however, is generally still nominal or ordinal. </li></ul>
  3. 3. ANOVA tells political scientists … <ul><li>(1) if there are any differences among the means </li></ul><ul><li>(2) which specific means differ and by how much </li></ul><ul><li>(3) whether the observed differences in Y could have arisen by chance or whether they reflect real variation among the categories or groups in X </li></ul>
  4. 4. Two important concepts … <ul><li>Effect size —The difference between one mean and the other. </li></ul><ul><li>Difference of means test —The larger the difference of means, the more likely the difference is not due to chance and is instead due to a relationship between the independent and dependent variables. </li></ul>
  5. 5. Setting up an example <ul><li>Suppose you want to test the effect of negative political ads on intention of voting in the next election. </li></ul><ul><li>You set up a control group and a test group. Each group watches a newscast, but the test group watches negative TV ads during the commercial breaks. The control group watches a newscast without a campaign ad. </li></ul><ul><li>You create a pre- and post-test of both groups to compare the effects of both ads. </li></ul>
  6. 6. Difference of the Means <ul><li>Effect = Mean (test group) – Mean (control group) </li></ul>
  7. 7. Difference of the Means <ul><li>Although different statistics use different formulas, each means test has two identical properties: </li></ul><ul><li>(1) The numerator indicates the difference of the means </li></ul><ul><li>(2) The denominator indicates the standard error of the difference of the means. </li></ul>
  8. 8. Difference of the Means <ul><li>A means test will compare the means of two </li></ul><ul><li>different samples. </li></ul><ul><li>The larger the N for both samples, the greater </li></ul><ul><li>confidence that the observed difference in the </li></ul><ul><li>sample (D) will correctly estimate the </li></ul><ul><li>population difference (∆). </li></ul>
  9. 9. Difference of the Means <ul><li>Difference of the means tests the null hypothesis that there is no difference between the means. </li></ul><ul><li>You can basically test the significance of a means difference by employing a hypothesis test or calculating a confidence interval. </li></ul>
  10. 10. Difference of the Means <ul><li>Logic is the same as for earlier z and t tests </li></ul><ul><li>Estimated difference of the means divided by the estimated standard error </li></ul><ul><li>z is for large samples </li></ul>
  11. 11. Large Sample Difference of Means <ul><li>Comparing the mean Obama thermometer score for men and women: </li></ul><ul><li>Men: Mean (45), Sample Size (400), Variance (10.2) </li></ul><ul><li>Women: Mean (55), Sample Size (500), Variance (11.4) </li></ul><ul><li>H 0 : There is no difference between the two samples. </li></ul><ul><li>H A : µ women - µ men > 0; α = .01, one-tail test </li></ul>
  12. 12. Difference of the Means <ul><li>55-45 = </li></ul><ul><li>√ (10.2/400 + 11.4/500) </li></ul><ul><li> 10 = </li></ul><ul><li>√ (.0255 + .0228) </li></ul><ul><li> 10 = </li></ul><ul><li>.220 </li></ul><ul><li>Z obs = 45.454 </li></ul>
  13. 13. Difference of the Means <ul><li>If | Z obs | ≥ Z crit , reject H 0 in favor of H A </li></ul><ul><li>| Z obs | = 45.454 > Z crit (2.325) </li></ul><ul><li>So, we reject H 0 in favor of H A </li></ul>