Sampling Distributions

Sampling Distribution Is the Theoretical probability distribution of a sample statistic Is the Theoretical probability distribution of a sample statistic A sample statistic is a random variable: A sample statistic is a random variable: E.g. Sample mean, sample proportion E.g. Sample mean, sample proportion For the mean: For the mean: It shows how sample means are distributed in relation to the true population mean It shows how sample means are distributed in relation to the true population mean

Why Study Sampling Distributions? Sample statistics are used to estimate population parameters Sample statistics are used to estimate population parameters e.g.: estimates the population mean e.g.: estimates the population mean Problems: Different samples provide different estimates Problems: Different samples provide different estimates Large samples give better estimates; large sample costs more Large samples give better estimates; large sample costs more How good is the estimate? How good is the estimate? Approach: Sampling distribution tells us how close our estimate should be to the true value Approach: Sampling distribution tells us how close our estimate should be to the true value

Developing Sampling Distributions Simple Example Case: Simple Example Case: Assume there is a population … Assume there is a population … Population size N=4 Population size N=4 Random variable, X, is age of individuals Random variable, X, is age of individuals Values of X: 18, 20, 22, 24 measured in years Values of X: 18, 20, 22, 24 measured in years A B C D

A B C D (18) (20) (22) (24) Uniform Distribution P(X) X Developing Sampling Distributions (continued) Summary Measures for the Population Distribution

Instead of considering the population of ages, let’s look at the SAMPLE MEANS All Possible Samples of Size n=2 16 Different Samples Can Be Taken 16 Sample Means

Sampling Distribution of All Sample Means P(X) X Sample Means Distribution 16 Sample Means _ Developing Sampling Distributions (continued)

Summary Measures of Sampling Distribution (continued) Mean of Sample Means Standard Deviation of Sample Means Developing Sampling Distributions

Comparing the Population with its Sampling Distribution P(X) X Sample Means Distribution n = 2 A B C D (18) (20) (22) (24) Population N = 4 P(X) X _

When the Population is Normal Central Tendency Variation Sampling with Replacement Population Distribution Sampling Distributions

When the Population is Not Normal Population Distribution Sampling Distributions Central Tendency Variation Sampling with Replacement

Central Limit Theorem As Sample Size Gets “Large Enough” Sampling Distribution Becomes Almost Normal Regardless of Shape of Population

Population Proportions Categorical variable Categorical variable e.g.: Gender, voted for bush, college degree e.g.: Gender, voted for bush, college degree Proportion of population that has a characteristic Proportion of population that has a characteristic Sample proportion provides an estimate Sample proportion provides an estimate If two outcomes, X has a binomial distribution If two outcomes, X has a binomial distribution Possess or do not possess characteristic Possess or do not possess characteristic

Sampling Distribution of a Sample Proportion Approximated by normal distribution if: Approximated by normal distribution if: and and Mean of samples: Mean of samples: Standard error of proportion : Standard error of proportion : p = population proportion p = population proportion Sampling Distribution P(p s ) psps

Standardizing Sampling Distribution of Proportion Sampling Distribution Standardized Normal Distribution

Example Sampling Distribution Standardized Normal Distribution

Election Polling Example: In the Gore/Bush Election, the proportion voting for Bush was practically p=0.5. If ABC News took a poll of 1067 adults, what is the probability that the sample proportion (p is within.03 of the population proportion? In the Gore/Bush Election, the proportion voting for Bush was practically p=0.5. If ABC News took a poll of 1067 adults, what is the probability that the sample proportion (p is within.03 of the population proportion?

We are interested in designing a study to estimate a given population parameter (MEAN) with certain precision. Estimate mean weight of 2576 babies born in the hospital with a 99% CI. Estimate of  = 1. You need to weight at least 27 babies to obtain an estimate to be 99% confident that the error will be  0.5 pounds You need to weight at least 16 babies to obtain an estimate to be 95% confident that the error will be  0.5 pounds

We are interested in designing a study to estimate a given population parameter (PROPORTION) with certain precision Determine proportion of adults living with hepatitis B virus. N required to estimate it within 0.03 with 95% confidence. In a similar area this proportion is What should be n if such estimate is not available? You need to recruit at least 683 people to obtain an estimate within 0.03 being 95% confident You need to recruit at most 1068 people to obtain an estimate within 0.03 being 95% confident -> p=0.5 = Worst case scenario! dz n zpp d   ()1 2 2

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