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PPA 415 Research Methods in Public
Administration

• Lecture 5 Normal Curve, Sampling, and Estimation

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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.

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Normal Curve
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Normal Curve
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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.

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Computing Z-Scores Fair Housing Survey 2000
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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 0.0026.
• From Appendix A, Table A Z0 is .5.
• P6-12.9 .5-.0026 .4974.
• 49.74 of the distribution lies between 6 and
  12.9 years of education

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Computing Z-Scores Examples

• What percentage of the cases are less than eight
  years of education?
• What percentage have more than 13 years?

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Computing Z-Scores Examples

• What percentage of Birmingham residents have
  between 10 and 13 years of education?

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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.

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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.

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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).

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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The Sampling Distribution

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

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The Sampling Distribution

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

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The Sampling Distribution
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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.

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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.

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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 a.
  If an infinite number of confidence intervals
  are constructed, 95 will contain the population
  mean and 5 wont.

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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.

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Estimation Procedures
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Estimation Procedures Sample Mean
Only use if sample is 100 or greater
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Estimation Proportions Large Sample
Use only if sample size is greater than 100
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Estimation Procedures

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

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Confidence Interval Examples
Birmingham Fair Housing Survey Education with
95, 99, and 99.9 confidence intervals.
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Confidence Interval Examples

• Proportion of sample who believe that
  discrimination is a major problem in Birmingham.