PPA 415 – Research Methods in Public Administration Lecture 5 – Normal Curve, Sampling, 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 – Fair Housing Survey 2000

Computing Z-Scores: Examples What percentage of the cases have between six and the mean years of education? From Appendix A, Table A: Z=-2.81 is From Appendix A, Table A: Z=0 is.5. P = = % of the distribution lies between 6 and 12.9 years of education

Computing Z-Scores: Examples What percentage of the cases are less than eight years of education? What percentage have more than 13 years?

Computing Z-Scores: Examples What percentage of Birmingham residents have between 10 and 13 years of education?

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 in Appendix A, Table A. If the distance is more than a score higher than the mean), subtract the probability in Appendix A, Table A 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.

Probability One interpretation of the area under the normal curve is as probabilities. Probabilities are determined as the number of successful events divided by the total possible number of events. The probability of selecting a king of hearts from a deck of cards is 1/52 or.0192 (1.92%).

Probability The proportions under the normal curve can be treated as probabilities that a randomly selected case will fall within the prescribed limits. Thus, in the Birmingham fair housing survey, the probability of selecting a resident with between 10 and 13 years of education is 39.7%.

Sampling One of the goals of social science research is to test our theories and hypotheses using many different types of people drawn from a broad cross section of society. However, the populations we are interested in are usually too large to test.

Sampling To deal with this problem, researchers select samples or subsets of the population. The goal is to learn about the populations using the data from the samples.

Sampling Basic procedures for selecting probability samples, the only kind that allow generalization to the larger population. Researcher do use nonprobability samples, but generalizing from them is nearly impossible. The goal of sampling is to select cases in the final sample that are representative of the population from which they are drawn. A sample is representative if it reproduces the important characteristics of the population.

Sampling The fundamental principle of probability sampling is that a sample is very likely to be representative if it is selected by the Equal Probability of Selection Method (EPSEM). Every case in the population must have an equal chance of ending up in the sample.

Sampling EPSEM and representativeness are not the same thing. EPSEM samples can be unrepresentative, but the probability of such an event can be calculated unlike nonprobability samples.

EPSEM Sampling Techniques Simple random sample – list of cases and a system for selection that ensures EPSEM. Systematic sampling – only the first case is randomly sample, then a skip interval is used. Stratified sample – random subsamples on the basis of some important characteristic. Cluster sampling – used when no list exists. Clusters often based on geography.

The Sampling Distribution Once we have selected a probability sample according to some EPSEM procedure, what do we know? We know a great deal about the sample, but nothing about the population. Somehow, we have to get from the sample to the population. The instrument used is the sampling distribution.

The Sampling Distribution The theoretical, probabilistic distribution of a descriptive statistic (such as the mean) for all possible samples of certain sample size (N). Three distributions are involved in every application of inferential statistics. The sample distribution – empirical, shape, central tendency and distribution. The population distribution – empirical, unknown. The sampling distribution – theoretical, shape, central tendency, and dispersion can be deduced.

The Sampling Distribution The sampling distribution allows us to estimate the probability of any sample outcome. Discuss the identification of a sampling distribution. Generally speaking, a sampling distribution will be symmetrical, approximately normal, and have the mean of the population.

The Sampling Distribution If repeated random samples of size N are drawn from a normal population with mean μ and standard deviation σ, then the sampling distribution of sample means will be normal with a mean μ and a standard deviation of σ/  N (standard error of the mean).

The Sampling Distribution Central Limit Theorem. If repeated random samples of size N are drawn from any population, with mean μ and standard deviation σ, then, as N becomes large, the sampling distribution of sample means will approach normality, with mean μ and standard deviation σ/  N. The theorem removes normality constraint in population. Rule of thumb: N  100.

The Sampling Distribution

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 – Proportions Large Sample Use only if sample size is greater than 100

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 Birmingham Fair Housing Survey Education with 95%, 99%, and 99.9% confidence intervals.

Confidence Interval Examples Proportion of sample who believe that discrimination is a major problem in Birmingham.