1 / 77

PPA 501 – Analytical Methods in Administration

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.

jerry
Download Presentation

PPA 501 – Analytical Methods in Administration

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

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

  2. 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.

  3. Normal Curve

  4. 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.

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

  6. 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 = -2.4878; =normsdist(-2.4878); p=0.006427 • From Excel, =standardize(0.0,0.01,0.205)= Z = -0.04878; =normsdist(-0.04878); p=0.480547 • P-0.5&0.0 = 0.480547-0.006427 = 0.474120. • 47.4% of the distribution lies between -0.56 and 0 on the ideology scale.

  7. Computing Z-Scores: Examples • What percentage of the House delegations from 1953 to 2005 have more conservative scores than 0.5? (1 - .992 = 0.008 or 0.8%) • What percentage have more liberal scores than -0.25? (10.2%).

  8. 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.

  9. 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.

  10. 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.

  11. 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.

  12. 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.

  13. 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.

  14. Estimation Procedures

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

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

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

  18. PPA 501 – Analytical Methods in Administration Lecture 6b – One-Sample and Two-Sample Tests

  19. 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.

  20. 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:

  21. 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 (one-tailed).

  22. 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.

  23. 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.

  24. Five-Step Model: Critical Choices

  25. 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?

  26. 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:

  27. 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)

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

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

  30. 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:

  31. 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.

  32. 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.

  33. 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).

  34. 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:

  35. 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.

  36. 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.

  37. 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.

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

  39. 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:

  40. 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).

  41. 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.

  42. 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?

  43. 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:

  44. 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).

  45. 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.

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

  47. 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:

  48. 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 A, Table B.

  49. 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.

  50. 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?

More Related