PPA 415

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.
• ?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.
• a0.05.
• 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.
• ?0.05.
• DfN-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.
• ?0.05.
• Df41-140.
• 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) gt 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.
• ?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.
• ?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.
• ?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.
• ?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 lt 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.
• ?0.05.
• DfN1N2-2
• 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.
• ?0.05, two-tailed.
• DfN1N2-21425-237
• 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.
• ?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.
• ?0.05.
• Z(critical)?1.96 (two-tailed).

51
Five-step Model Large Two-sample Proportions
(Example)

• Step 4. Computing the test statistic.
• Step 5. Making a decision.
• Z(obtained) is greater than z(critical),
  therefore reject the null hypothesis that the two
  administrations have the same major disaster
  declaration percentages. The two presidential
  administrations have different approval rates.