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. 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.

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:

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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).

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.

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.

Five-Step Model: Critical Choices

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?

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:

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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)

Five-step Model of Hypothesis Testing – Large-sample Z Scores

Step 4. Computing the test statistic.

Step 5. Making a decision.

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Five-Step Model: Small Sample T-test (One Sample)

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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:

1

testtailed-one;

or test tailed-one;

testtailed-two;

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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.

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.

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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).

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:

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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.

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.

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Five Step Model: Large Sample Proportions.

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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:

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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).

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.

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Five Step Model: Large Sample Proportions.

Do residents of Birmingham, Alabama, have significantly different homeownership rates than all residents of the United States?

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:

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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).

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.

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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.

Two-Sample Models – Large Samples

Formula.

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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:

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testtailed-one;

or test tailed-one;

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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).

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.

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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?

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:

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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).

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.

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Five-Step Model – Small Two-Sample Tests

If N1 + N2 < 100, use this formula.

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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:

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or test tailed-one;

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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.

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.

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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?

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:

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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

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.

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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:

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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).

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.

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Five-Step Model – Large Two-Sample Tests (Proportions)

Did Presidents Ford and Carter have different approval rates for major disaster declarations?

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:

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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).

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.

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