POLS 7000X STATISTICS IN POLITICAL SCIENCE CLASS 5 BROOKLYN COLLEGE-CUNY SHANG E. HA Leon-Guerrero and Frankfort-Nachmias, Essentials of Statistics for a Diverse Society

Leon-Guerrero/Frankfort-Nachmias: Essentials of Social Statistics for a Diverse Society © 2012 SAGE Publications Chapter 6: Sampling, Sampling Distributions, and Estimation  Aims of Sampling  Probability Sampling  The Concept of the Sampling Distribution  The Sampling Distribution of the Mean  The Central Limit Theorem  Estimation  Procedures for Estimating Confidence Intervals  Confidence Intervals for Proportions  Statistics in Practice: Health Care Reform  Statistics in Practice: The Margin of Error

Leon-Guerrero/Frankfort-Nachmias: Essentials of Social Statistics for a Diverse Society © 2012 SAGE Publications Sampling  Population – A group that includes all the cases (individuals, objects, or groups) in which the researcher is interested.  Sample – A relatively small subset from a population.

Leon-Guerrero/Frankfort-Nachmias: Essentials of Social Statistics for a Diverse Society © 2012 SAGE Publications Notation

Leon-Guerrero/Frankfort-Nachmias: Essentials of Social Statistics for a Diverse Society © 2012 SAGE Publications Sampling  Parameter – A measure (for example, mean or standard deviation) used to describe a population distribution.  Statistic – A measure (for example, mean or standard deviation) used to describe a sample distribution.

Leon-Guerrero/Frankfort-Nachmias: Essentials of Social Statistics for a Diverse Society © 2012 SAGE Publications Sampling: Parameter & Statistic

Leon-Guerrero/Frankfort-Nachmias: Essentials of Social Statistics for a Diverse Society © 2012 SAGE Publications Probability Sampling  Probability sampling – A method of sampling that enables the researcher to specify for each case in the population the probability of its inclusion in the sample.

Leon-Guerrero/Frankfort-Nachmias: Essentials of Social Statistics for a Diverse Society © 2012 SAGE Publications Random Sampling  Simple Random Sample – A sample designed in such a way as to ensure that (1) every member of the population has an equal chance of being chosen and (2) every combination of N members has an equal chance of being chosen.  This can be done using a computer, calculator, or a table of random numbers

Population inferences can be made...

...by selecting a representative sample from the population

Leon-Guerrero/Frankfort-Nachmias: Essentials of Social Statistics for a Diverse Society © 2012 SAGE Publications Random Sampling  Systematic random sampling – A method of sampling in which every Kth member in the total population is chosen for inclusion in the sample after the first member of the sample is selected at random from among the first K members of the population.  Where

Leon-Guerrero/Frankfort-Nachmias: Essentials of Social Statistics for a Diverse Society © 2012 SAGE Publications Sampling Distributions  Sampling error – The discrepancy between a sample estimate of a population parameter and the real population parameter.  Sampling distribution – A theoretical distribution of all possible sample values for the statistic in which we are interested.

Leon-Guerrero/Frankfort-Nachmias: Essentials of Social Statistics for a Diverse Society © 2012 SAGE Publications  Sampling distribution of the mean – A theoretical probability distribution of sample means that would be obtained by drawing from the population all possible samples of the same size. If we repeatedly drew samples from a population and calculated the sample means, those sample means would be normally distributed (as the number of samples drawn increases.) The next several slides demonstrate this.  Standard error of the mean – The standard deviation of the sampling distribution of the mean. It describes how much dispersion there is in the sampling distribution of the mean. Sampling Distributions

Leon-Guerrero/Frankfort-Nachmias: Essentials of Social Statistics for a Diverse Society © 2012 SAGE Publications If all possible random samples of size N are drawn from a population with mean  y and a standard deviation, then as N becomes larger, the sampling distribution of sample means becomes approximately normal, with mean and standard deviation. The Central Limit Theorem

Leon-Guerrero/Frankfort-Nachmias: Essentials of Social Statistics for a Diverse Society © 2012 SAGE Publications Distribution of Sample Means with 21 Samples Sample Means S.D. = 2.02 Mean of means = 41.0 Number of Means = 21 Frequency Distribution of Sample Means with 21 Samples

Leon-Guerrero/Frankfort-Nachmias: Essentials of Social Statistics for a Diverse Society © 2012 SAGE Publications Distribution of Sample Means with 96 Samples Frequency Sample Means S.D. = 1.80 Mean of Means = Number of Means = 96

Leon-Guerrero/Frankfort-Nachmias: Essentials of Social Statistics for a Diverse Society © 2012 SAGE Publications Distribution of Sample Means with 170 Samples Frequency Sample Means S.D. = 1.71 Mean of Means= Number of Means= 170

Leon-Guerrero/Frankfort-Nachmias: Essentials of Social Statistics for a Diverse Society © 2012 SAGE Publications Estimation Defined:  Estimation – A process whereby we select a random sample from a population and use a sample statistic to estimate a population parameter.

Leon-Guerrero/Frankfort-Nachmias: Essentials of Social Statistics for a Diverse Society © 2012 SAGE Publications Point and Interval Estimation  Point Estimate – A sample statistic used to estimate the exact value of a population parameter  Confidence interval (interval estimate) – A range of values defined by the confidence level within which the population parameter is estimated to fall.  Confidence Level – The likelihood, expressed as a percentage or a probability, that a specified interval will contain the population parameter.

Leon-Guerrero/Frankfort-Nachmias: Essentials of Social Statistics for a Diverse Society © 2012 SAGE Publications Take a subset of the population Estimations Lead to Inferences

Leon-Guerrero/Frankfort-Nachmias: Essentials of Social Statistics for a Diverse Society © 2012 SAGE Publications Try and reach conclusions about the population Estimations Lead to Inferences

Leon-Guerrero/Frankfort-Nachmias: Essentials of Social Statistics for a Diverse Society © 2012 SAGE Publications A population distribution – variation in the larger group that we want to know about. A distribution of sample observations – variation in the sample that we can observe. A sampling distribution – a normal distribution whose mean and standard deviation are unbiased estimates of the parameters and allows one to infer the parameters from the statistics. Inferential Statistics Involves Three Distributions:

Leon-Guerrero/Frankfort-Nachmias: Essentials of Social Statistics for a Diverse Society © 2012 SAGE Publications What does this Theorem tell us: –Even if a population distribution is skewed, we know that the sampling distribution of the mean is normally distributed –As the sample size gets larger, the mean of the sampling distribution becomes equal to the population mean –As the sample size gets larger, the standard error of the mean decreases in size (which means that the variability in the sample estimates from sample to sample decreases as N increases). It is important to remember that researchers do not typically conduct repeated samples of the same population. Instead, they use the knowledge of theoretical sampling distributions to construct confidence intervals around estimates. The Central Limit Theorem Revisited