CHAPTER 6: SAMPLING, SAMPLING DISTRIBUTIONS, AND ESTIMATION 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

Leon-Guerrero/Frankfort-Nachmias: Essentials of Social Statistics for a Diverse Society © 2012 SAGE Publications Confidence Levels:  Confidence Level – The likelihood, expressed as a percentage or a probability, that a specified interval will contain the population parameter.  95% confidence level – there is a.95 probability that a specified interval DOES contain the population mean. In other words, there are 5 chances out of 100 (or 1 chance out of 20) that the interval DOES NOT contain the population mean.  99% confidence level – there is 1 chance out of 100 that the interval DOES NOT contain the population mean.

Leon-Guerrero/Frankfort-Nachmias: Essentials of Social Statistics for a Diverse Society © 2012 SAGE Publications Constructing a Confidence Interval (CI)  The sample mean is the point estimate of the population mean.  The sample standard deviation is the point estimate of the population standard deviation.  The standard error of the mean makes it possible to state the probability that an interval around the point estimate contains the actual population mean.

Leon-Guerrero/Frankfort-Nachmias: Essentials of Social Statistics for a Diverse Society © 2012 SAGE Publications What We are Wanting to Do We want to construct an estimate of where the population mean falls based on our sample statistics This is our Confidence Interval The actual population parameter falls somewhere on this line

Leon-Guerrero/Frankfort-Nachmias: Essentials of Social Statistics for a Diverse Society © 2012 SAGE Publications Standard error of the mean – the standard deviation of a sampling distribution Standard Error The Standard Error

Leon-Guerrero/Frankfort-Nachmias: Essentials of Social Statistics for a Diverse Society © 2012 SAGE Publications Since the standard error is generally not known, we usually work with the estimated standard error: Estimating Standard Errors

Leon-Guerrero/Frankfort-Nachmias: Essentials of Social Statistics for a Diverse Society © 2012 SAGE Publications where: =sample mean (estimate of  ) Z=Z score for one-half the acceptable error =estimated standard error Determining a Confidence Interval (CI)

Leon-Guerrero/Frankfort-Nachmias: Essentials of Social Statistics for a Diverse Society © 2012 SAGE Publications Confidence Level – Increasing our confidence level from 95% to 99% means we are less willing to draw the wrong conclusion – we take a 1% risk (rather than a 5%) that the specified interval does not contain the true population mean. If we reduce our risk of being wrong, then we need a wider range of values... So the interval becomes less precise. Confidence Interval Width

Leon-Guerrero/Frankfort-Nachmias: Essentials of Social Statistics for a Diverse Society © 2012 SAGE Publications Confidence Interval Width More confident, less precise More precise, less confident

Leon-Guerrero/Frankfort-Nachmias: Essentials of Social Statistics for a Diverse Society © 2012 SAGE Publications Confidence Interval Z Values

Leon-Guerrero/Frankfort-Nachmias: Essentials of Social Statistics for a Diverse Society © 2012 SAGE Publications Sample Size – Larger samples result in smaller standard errors, and therefore, in sampling distributions that are more clustered around the population mean. A more closely clustered sampling distribution indicates that our confidence intervals will be narrower and more precise. Confidence Interval Width

Leon-Guerrero/Frankfort-Nachmias: Essentials of Social Statistics for a Diverse Society © 2012 SAGE Publications Standard Deviation – Smaller sample standard deviations result in smaller, more precise confidence intervals. (Unlike sample size and confidence level, the researcher plays no role in determining the standard deviation of a sample.) Confidence Interval Width

Leon-Guerrero/Frankfort-Nachmias: Essentials of Social Statistics for a Diverse Society © 2012 SAGE Publications Example: Sample Size and Confidence Intervals

Leon-Guerrero/Frankfort-Nachmias: Essentials of Social Statistics for a Diverse Society © 2012 SAGE Publications Estimating the standard error of a proportion – based on the Central Limit Theorem, a sampling distribution of proportions is approximately normal, with a mean,  p, equal to the population proportion, , and with a standard error of proportions equal to: Since the standard error of proportions is generally not known, we usually work with the estimated standard error: Confidence Intervals for Proportions

Leon-Guerrero/Frankfort-Nachmias: Essentials of Social Statistics for a Diverse Society © 2012 SAGE Publications where: p=observed sample proportion (estimate of  ) Z=Z score for one-half the acceptable error s p =estimated standard error of the proportion Determining a Confidence Interval for a Proportion

Leon-Guerrero/Frankfort-Nachmias: Essentials of Social Statistics for a Diverse Society © 2012 SAGE Publications Confidence Intervals for Proportions Protestants in favor of banning stem cell research: N = 2,188, p =.37 Calculate the estimated standard error:.01 Determine the confidence level = (.010) =.37 ±.020 =.35 to.39 Lets say we want to be 95% confident

Leon-Guerrero/Frankfort-Nachmias: Essentials of Social Statistics for a Diverse Society © 2012 SAGE Publications Confidence Intervals for Proportions Catholics in favor of banning stem cell research: N = 880, p =.32 Calculate the estimated standard error:.016 Determine the confidence level = (.016) =.32 ±.031 =.29 to.35 Lets say we want to be 95% confident

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