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PPA 415 – Research Methods in Public Administration

PPA 415 – Research Methods in Public Administration. Lecture 6 – One-Sample and Two-Sample Tests. Five-step Model of Hypothesis Testing. Step 1. Making assumptions and meeting test requirements. Step 2. Stating the null hypothesis.

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PPA 415 – Research Methods in Public Administration

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  1. PPA 415 – Research Methods in Public Administration Lecture 6 – One-Sample and Two-Sample Tests

  2. Five-step Model of Hypothesis Testing • Step 1. Making assumptions and meeting test requirements. • Step 2. Stating the null hypothesis. • Step 3. Selecting the sampling distribution and establishing the critical region. • Step 4. Computing the test statistic. • Step 5. Making a decision and interpreting the results of the test.

  3. Five-step Model of Hypothesis Testing – One-sample Z Scores • Step 1. Making assumptions. • Model: random sampling. • Interval-ratio measurement. • Normal sampling distribution. • Step 2. Stating the null hypothesis (no difference) and the research hypothesis. • Ho: • H1:

  4. Five-step Model of Hypothesis Testing – One-sample Z Scores • Step 3. Selecting the sampling distribution and establishing the critical region. • Sampling distribution = Z distribution. • Α=0.05. • Z(critical)=1.96 (two-tailed); +1.65 or -1.65 (two-tailed).

  5. Five-step Model of Hypothesis Testing – One-sample Z Scores • Step 4. Computing the test statistic. • Use z-formula. • Step 5. Making a decision. • Compare z-critical to z-obtained. If z-obtained is greater in magnitude than z-critical, reject null hypothesis. Otherwise, accept null hypothesis.

  6. Five-Step Model: Critical Choices • Choice of alpha level: .05, .01, .001. • Selection of research hypothesis. • Two-tailed test: research hypothesis simplify states that means of sample and population are different. • One-tailed test: mean of sample is larger or smaller than mean of population. • Type of error to maximize: Type I or Type II. • Type I – rejecting a null hypothesis that is true. • Type II – accepting a null hypothesis that is false.

  7. Five-Step Model: Critical Choices

  8. Five-step Model: Example • Is the average age of voters in the 2000 National Election Study different than the average age of all adults in the U.S. population?

  9. Five-step Model of Hypothesis Testing – Large-sample Z Scores • Step 1. Making assumptions. • Model: random sampling. • Interval-ratio measurement. • Normal sampling distribution. • Step 2. Stating the null hypothesis (no difference) and the research hypothesis. • Ho: • H1:

  10. Five-step Model of Hypothesis Testing – Large-sample Z Scores • Step 3. Selecting the sampling distribution and establishing the critical region. • Sampling distribution = Z distribution. • α=0.05. • Z(critical)=1.96 (two-tailed)

  11. Five-step Model of Hypothesis Testing – Large-sample Z Scores • Step 4. Computing the test statistic. • Step 5. Making a decision.

  12. Five-Step Model: Small Sample T-test (One Sample) • Formula

  13. Five-Step Model: Small Sample T-test (One Sample) • Step 1. Making Assumptions. • Random sampling. • Interval-ratio measurement. • Normal sampling distribution. • Step 2. Stating the null hypothesis. • Ho: • H1:

  14. Five-step Model of Hypothesis Testing – One-sample Z Scores • Step 3. Selecting the sampling distribution and establishing the critical region. • Sampling distribution = t distribution. • Α=0.05. • Df=N-1. • t(critical) from Appendix B, p. 359 in Healey.

  15. Five-step Model of Hypothesis Testing – One-sample Z Scores • Step 4. Computing the test statistic. • Step 5. Making a decision. • Compare t-critical to t-obtained. If t-obtained is greater in magnitude than t-critical, reject null hypothesis. Otherwise, accept null hypothesis.

  16. Five-step Model of Hypothesis Testing – One-sample Z Scores • Is the average age of individuals in the JCHA 2000 sample survey older than the national average age for all adults? (One-tailed).

  17. Five-Step Model: Small Sample T-test (One Sample) – JCHA 2000 • Step 1. Making Assumptions. • Random sampling. • Interval-ratio measurement. • Normal sampling distribution. • Step 2. Stating the null hypothesis. • Ho: • H1:

  18. Five-Step Model: Small Sample T-test (One Sample) – JCHA 2000 • Step 3. Selecting the sampling distribution and establishing the critical region. • Sampling distribution = t distribution. • Α=0.05. • Df=41-1=40. • t(critical) =1.684.

  19. Five-Step Model: Small Sample T-test (One Sample) – JCHA 2000 • Step 4. Computing the test statistic. • Step 5. Making a decision. • T(obtained) > t(critical). Therefore, reject the null hypothesis. The sample of residents from the Jefferson County Housing Authority is significantly older than the adult population of the United States.

  20. Five Step Model: Large Sample Proportions. • Formula.

  21. Five Step Model: Large Sample Proportions • Step 1. Making assumptions. • Model: random sampling. • Nominal measurement. • Normal shaped sampling distribution. • Step 2. Stating the null hypothesis (no difference) and the research hypothesis. • Ho: • H1:

  22. Five Step Model: Large Sample Proportions. • Step 3. Selecting the sampling distribution and establishing the critical region. • Sampling distribution = Z distribution. • Α=0.05, one or two-tailed. • Z(critical)=1.96 (two-tailed); +1.65 or -1.65 (two-tailed).

  23. Five Step Model: Large Sample Proportions. • Step 4. Computing the test statistic. • Step 5. Making a decision. • Compare z-critical to z-obtained. If z-obtained is greater in magnitude than z-critical, reject null hypothesis. Otherwise, accept null hypothesis.

  24. Five Step Model: Large Sample Proportions. • Do residents of Birmingham, Alabama, have significantly different homeownership rates than all residents of the United States?

  25. Five Step Model: Large Sample Proportions. Homeownership in Birmingham, Alabama • Step 1. Making assumptions. • Model: random sampling. • Nominal measurement. • Normal shaped sampling distribution. • Step 2. Stating the null hypothesis (no difference) and the research hypothesis. • Ho: • H1:

  26. Five Step Model: Large Sample Proportions. • Step 3. Selecting the sampling distribution and establishing the critical region. • Sampling distribution = Z distribution. • Α=0.05, two-tailed. • Z(critical)=1.96 (two-tailed).

  27. Five Step Model: Large Sample Proportions. • Step 4. Computing the test statistic. • Step 5. Making a decision. • The absolute value of z-obtained is greater than the absolute value of Z-critical, therefore reject the null hypothesis. The homeownership rate in Birmingham is significantly different than the national rate.

  28. Two-Sample Models – Large Samples • Most of the time we do not have the population means or proportions. All we can do is compare the means or proportions of population subsamples. • Adds the additional assumption of independent random samples.

  29. Two-Sample Models – Large Samples • Formula.

  30. Five-Step Model – Large Two-Sample Tests (Z Distribution) • Step 1. Making assumptions. • Model: Independent random samples. • Interval-ratio measurement. • Normal sampling distribution. • Step 2. Stating the null hypothesis (no difference) and the research hypothesis. • Ho: • H1:

  31. Five-Step Model – Large Two-Sample Tests (Z Distribution) • Step 3. Selecting the sampling distribution and establishing the critical region. • Sampling distribution = Z distribution. • Α=0.05. • Z(critical)=1.96 (two-tailed); +1.65 or -1.65 (one-tailed).

  32. Five-Step Model – Large Two-Sample Tests (Z Distribution) • Step 4. Computing the test statistic. • Step 5. Making a decision. • Compare z-critical to z-obtained. If z-obtained is greater in magnitude than z-critical, reject null hypothesis. Otherwise, accept null hypothesis.

  33. Five-Step Model – Large Two-Sample Tests (Z Distribution) • Do non-white citizens of Birmingham, Alabama, believe that discrimination is more of a problem than white citizens?

  34. Five-Step Model – Large Two-Sample Tests (Fair Housing) • Step 1. Making assumptions. • Model: Independent random samples. • Interval-ratio measurement. • Normal sampling distribution. • Step 2. Stating the null hypothesis (no difference) and the research hypothesis. • Ho: • H1:

  35. Five-Step Model – Large Two-Sample Tests (Z Distribution) • Step 3. Selecting the sampling distribution and establishing the critical region. • Sampling distribution = Z distribution. • Α=0.05. • Z(critical)=+1.65 (one-tailed).

  36. Five-Step Model – Large Two-Sample Tests (Z Distribution) • Step 4. Computing the test statistic. • Step 5. Making a decision. • Z(obtained) is greater than Z(critical), therefore reject the null hypothesis of no difference. Non-whites believe that discrimination is more of a problem in Birmingham.

  37. Five-Step Model – Small Two-Sample Tests • If N1 + N2 < 100, use this formula.

  38. Five-Step Model – Small Two-Sample Tests (t Distribution) • Step 1. Making assumptions. • Model: Independent random samples. • Interval-ratio measurement. • Equal population variances • Normal sampling distribution. • Step 2. Stating the null hypothesis (no difference) and the research hypothesis. • Ho: • H1:

  39. Five-Step Model – Small Two-Sample Tests (t Distribution) • Step 3. Selecting the sampling distribution and establishing the critical region. • Sampling distribution = t distribution. • Α=0.05. • Df=N1+N2-2 • t(critical). See Appendix B, p. 359.

  40. Five-Step Model – Small Two-Sample Tests (t Distribution) • Step 4. Computing the test statistic. • Step 5. Making a decision. • Compare t-critical to t-obtained. If t-obtained is greater in magnitude than t-critical, reject null hypothesis. Otherwise, accept null hypothesis.

  41. Five-Step Model – Small Two-Sample Tests (t Distribution) • Did white and nonwhite residents of the Jefferson County Housing Authority have significantly different lengths of residence in 2000?

  42. Five-Step Model – Small Two-Sample Tests (JCHA 2000) • Step 1. Making assumptions. • Model: Independent random samples. • Interval-ratio measurement. • Equal population variances • Normal sampling distribution. • Step 2. Stating the null hypothesis (no difference) and the research hypothesis. • Ho: • H1:

  43. Five-Step Model – Small Two-Sample Tests (JCHA 2000) • Step 3. Selecting the sampling distribution and establishing the critical region. • Sampling distribution = t distribution. • Α=0.05, two-tailed. • Df=N1+N2-2=14+25-2=37 • t(critical) from Appendix B = 2.042

  44. Five-Step Model – Small Two-Sample Tests (t Distribution) • Step 4. Computing the test statistic. • Step 5. Making a decision. • Z(obtained) is less than Z(critical) in magnitude. Accept the null hypothesis. Whites and nonwhites in the JCHA 2000 survey do not have different lengths of residence in public housing.

  45. Five-Step Model – Large Two-Sample Tests (Proportions) • Step 1. Making assumptions. • Model: Independent random samples. • Interval-ratio measurement. • Normal sampling distribution. • Step 2. Stating the null hypothesis (no difference) and the research hypothesis. • Ho: • H1:

  46. Five-Step Model – Large Two-Sample Tests (Proportions) • Step 3. Selecting the sampling distribution and establishing the critical region. • Sampling distribution = Z distribution. • Α=0.05. • Z(critical)=1.96 (two-tailed); +1.65 or -1.65 (one-tailed).

  47. Five-Step Model – Large Two-Sample Tests (Proportions) • Step 4. Computing the test statistic. • Step 5. Making a decision. • Compare z-critical to z-obtained. If z-obtained is greater in magnitude than z-critical, reject null hypothesis. Otherwise, accept null hypothesis.

  48. Five-Step Model – Large Two-Sample Tests (Proportions) • Did Presidents Ford and Carter have different approval rates for major disaster declarations?

  49. Five-Step Model – Large Two-Sample Proportions (Example) • Step 1. Making assumptions. • Model: Independent random samples. • Interval-ratio measurement. • Normal sampling distribution. • Step 2. Stating the null hypothesis (no difference) and the research hypothesis. • Ho: • H1:

  50. Five-Step Model – Large Two-Sample Proportions (Example) • Step 3. Selecting the sampling distribution and establishing the critical region. • Sampling distribution = Z distribution. • Α=0.05. • Z(critical)=1.96 (two-tailed).

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