Sampling Distributions and Inference for Proportions(C18-C22 BVD) C18: Sampling Distributions

* The sampling distribution of a statistic is the distribution of values taken by the statistic in all possible samples of the same size from the same population. * Parameter – number that describes a characteristic of a population (like a mean or proportion) – use Greek letters to denote. * Statistic – number that describes a characteristic of a sample, often used to estimate to a parameter. * Sampling Variability – If you repeatedly sample from a population, the statistic you measure varies from sample to sample even though the parameter is constant.

* Population Distribution – a graph that shows how a characteristic is represented in a population (for example, 30% red, 70% blue) Maybe a bar graph. * Usually we don’t know the population parameter distribution, so we take a sample. The sample might be 25% red and 75% blue. We could graph this as a sample distribution. Bar Graph. * We could then graph the proportion of red on a number line, take a new sample, graph that, and do that over and over until we’ve gotten every possible sample. That dotplot or histogram would be the sampling distribution for the proportion of red in the population.

* A statistic is an unbiased estimator of a parameter if the mean of its sampling distribution if equal to the true value of the parameter being estimated. * Means and proportions of good samples are unbiased estimators, so making a sampling distribution can give us great information about the true population parameter. * This leads us to many powerful inference techniques for estimating and evaluating evidence against hypothesized population parameters.

* The variability of a sampling distribution is described by the spread of its sampling distribution. To decrease the spread, use a larger sample size in building the sampling distribution. * The ideal is low variability, low bias – i.e. large sample size, random sampling of an unbiased estimator.

* The sampling distribution for a proportion can be modeled by N(p, sqrt(pq/n)) * IF the following conditions are met: * 10%, Success/Fail (see Ch 17 show)

* According to the Central Limit Theorem (CLT), the sampling distribution for a mean can be modeled by * N(µ,σ/sqrt(n)) * If the following conditions are met: * Random sampling, independence, 10%

* The Standard Deviation of a sampling distribution for a proportion is sqrt(pq/n) but often we do not know p and q – population parameters. So, * We estimate them with p-hat and q-hat, and this makes our standard deviation an estimate, too. Since it is an estimate it is called a standard error. * The Standard Deviation of a sampling distribution for a mean is sigma/sqrt(n), but we rarely know sigma, so we estimate it with Sx, and then the standard deviation estimate is called a standard error.