PPA 501 – Analytical Methods in Administration Lecture 6a – Normal Curve, Z- Scores, and Estimation

Normal Curve The normal curve is central to the theory that underlies inferential statistics. The normal curve is a theoretical model. A frequency polygon that is perfectly symmetrical and smooth. Bell shaped, unimodal, with infinite tails. Crucial point distances along the horizontal axis, when measured in standard deviations, always measure the same proportion under the curve.

Normal Curve

Computing Z-Scores To find the percentage of the total area (or number of cases) above, below, or between scores in an empirical distribution, the original scores must be expressed in units of the standard deviation or converted into Z scores.

Computing Z-Scores – Mean ideology of House delegation by state.

Computing Z-Scores: Examples What percentage of the cases fall between -0.5 and 0.01 on the ideology scale? From Excel, =standardize(-0.5, 0.01, 0.205) = Z = ; =normsdist( ); p= From Excel, =standardize(0.0,0.01,0.205)= Z = ; =normsdist( ); p= P -0.5&0.0 = = % of the distribution lies between and 0 on the ideology scale.

Computing Z-Scores: Examples What percentage of the House delegations from 1953 to 2005 have more conservative scores than 0.5? ( = or 0.8%) What percentage have more liberal scores than -0.25? (10.2%).

Computing Z-scores: Rules If you want the distance between a score and the mean, subtract the probability from.5 if the Z is negative. Subtract.5 from the probability if Z is positive. If you want the distance beyond a score (less than a score lower than the mean), use the probability score from Excel. If the distance is more than a score higher than the mean), subtract the probability in Excel from 1.

Computing Z-scores: Rules If you want the difference between two scores other than the mean: Calculate Z for each score, identify the appropriate probability, and subtract the smaller probability from the larger.

Estimation Procedures Bias – does the mean of the sampling distribution equal the mean of the population? Efficiency – how closely around the mean does the sampling distribution cluster. You can improve efficiency by increasing sample size.

Estimation Procedures Point estimate – construct a sample, calculate a proportion or mean, and estimate the population will have the same value as the sample. Always some probability of error.

Estimation Procedures Confidence interval – range around the sample mean. First step: determine a confidence level: how much error are you willing to tolerate. The common standard is 5% or.05. You are willing to be wrong 5% of the time in estimating populations. This figure is known as alpha or α. If an infinite number of confidence intervals are constructed, 95% will contain the population mean and 5% won’t.

Estimation Procedures We now work in reverse on the normal curve. Divide the probability of error between the upper and lower tails of the curve (so that the 95% is in the middle), and estimate the Z-score that will contain 2.5% of the area under the curve on either end. That Z-score is ±1.96. Similar Z-scores for 90% (alpha=.10), 99% (alpha=.01), and 99.9% (alpha=.001) are ±1.65, ±2.58, and ±3.29.

Estimation Procedures

Estimation Procedures – Sample Mean Only use if sample is 100 or greater

Estimation Procedures You can control the width of the confidence intervals by adjusting the confidence level or alpha or by adjusting sample size.

Confidence Interval Examples Mean House Ideology for presidential disaster requests (1953 to 2005) with 90%, 95%, and 99% confidence intervals.

PPA 501 – Analytical Methods in Administration Lecture 6b – 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. H o : H 1 :

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); or (one-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. H o : H 1 :

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.

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

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. H o : H 1 :

Five-step Model of Hypothesis Testing – One-sample t 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 A, Table B in Agresti and Franklin.

Five-step Model of Hypothesis Testing – One-sample t 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.

Five-step Model of Hypothesis Testing – One-sample t 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. H o : H 1 :

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.

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.

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. H o : H 1 :

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); or (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.

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. H o : H 1 :

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.

Five-Step Model – Small Two- Sample Tests If N 1 + N 2 < 100, use this formula.

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. H o : H 1 :

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=N 1 +N 2 -2 t(critical). See Appendix A, Table B.

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.

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. H o : H 1 :

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=N 1 +N 2 -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.

PPA 501 – Analytical Methods in Administration Lecture 6c – Analysis of Variance

Introduction Analysis of variance (ANOVA) can be considered an extension of the t-test. The t-test assumes that the independent variable has only two categories. ANOVA assumes that the nominal or ordinal independent variable has two or more categories.

Introduction The null hypothesis is that the populations from which the each of samples (categories) are drawn are equal on the characteristic measured (usually a mean or proportion).

Introduction If the null hypothesis is correct, the means for the dependent variable within each category of the independent variable should be roughly equal. ANOVA proceeds by making comparisons across the categories of the independent variable.

Computation of ANOVA The computation of ANOVA compares the amount of variation within each category (SSW) to the amount of variation between categories (SSB). Total sum of squares.

Computation of ANOVA Sum of squares within (variation within categories). Sum of squares between (variation between categories).

Computation of ANOVA Degrees of freedom.

Computation of ANOVA Mean square estimates.

Computation of ANOVA Computational steps for shortcut. Find SST using computation formula. Find SSB. Find SSW by subtraction. Calculate degrees of freedom. Construct the mean square estimates. Compute the F-ratio.

Five-Step Hypothesis Test for ANOVA. Step 1. Making assumptions. Independent random samples. Interval ratio measurement. Normally distributed populations. Equal population variances. Step 2. Stating the null hypothesis.

Five-Step Hypothesis Test for ANOVA. Step 3. Selecting the sampling distribution and establishing the critical region. Sampling distribution = F distribution. Alpha =.05 (or.01 or...). Degrees of freedom within = N – k. Degrees of freedom between = k – 1. F-critical=Use Appendix D, p Step 4. Computing the test statistic. Use the procedure outlined above.

Five-Step Hypothesis Test for ANOVA. Step 5. Making a decision. If F(obtained) is greater than F(critical), reject the null hypothesis of no difference. At least one population mean is different from the others.

ANOVA – Example 1 – JCHA 2000 What impact does marital status have on respondent’s rating Of JCHA services? Sum of Rating Squared is 615

ANOVA – Example 1 – JCHA 2000 Step 1. Making assumptions. Independent random samples. Interval ratio measurement. Normally distributed populations. Equal population variances. Step 2. Stating the null hypothesis.

ANOVA – Example 1 – JCHA 2000 Step 3. Selecting the sampling distribution and establishing the critical region. Sampling distribution = F distribution. Alpha =.05. Degrees of freedom within = N – k = 38 – 5 = 33. Degrees of freedom between = k – 1 = 5 – 1 = 4. F-critical=2.69.

ANOVA – Example 1 – JCHA 2000 Step 4. Computing the test statistic.

ANOVA – Example 1 – JCHA 2000

ANOVA – Example 1 – JCHA Step 5. Making a decision. F(obtained) is F(critical) is F(obtained) < F(critical). Therefore, we fail to reject the null hypothesis of no difference. Approval of JCHA services does not vary significantly by marital status.

ANOVA – Example 2 – Presidential Disaster Set What impact does Presidential administration have on the president’s recommendation of disaster assistance?

ANOVA – Example 2 – Presidential Disaster Data Set Step 1. Making assumptions. Independent random samples. Interval ratio measurement. Normally distributed populations. Equal population variances. Step 2. Stating the null hypothesis.

ANOVA – Example 2 – Presidential Disaster Data Set Step 3. Selecting the sampling distribution and establishing the critical region. Sampling distribution = F distribution. Alpha =.05. Degrees of freedom within = N – k = 2642 – 10 = Degrees of freedom between = k – 1 = 10 – 1 = 9. F-critical=1.883.

ANOVA – Example 2 – Presidential Disaster Data Set Step 4. Computing the test statistic.

ANOVA – Example 2 – Presidential Disaster Data Set Step 5. Making a decision. F(obtained) is F(critical) is F(obtained) > F(critical). Therefore, we can reject the null hypothesis of no difference. Approval of federal disaster assistance does vary by presidential administration.