Introduction to the Practice of Statistics Fifth Edition Chapter 5: Sampling Distributions Copyright © 2005 by W. H. Freeman and Company David S. Moore George P. McCabe

Sampling Distributions 5.1 Sampling Distributions for Counts and Proportions 5.2 The Sampling Distribution of a Sample Mean

Basic Terminology The population distribution of a variable gives for a randomly chosen individual from the population, how likely the value of the variable for the individual is in certain ranges. Example: If the variable X (height of American women) is normal with mean, inches, and standard deviation, inches, then how likely is it that a randomly selected American women is over 65 inches tall?

More Terminology If we consider all SRSs of size n from the population of American women, what will the distribution of the sample means from each SRS?? This distribution is called the sampling distribution of a sample mean (from samples of size n). In this case, will denote the mean of the sampling distribution by and the standard deviation of the sampling distribution by. Via its shape, center, and spread, the sampling distribution of a statistic generally tells us how likely the statistic is to have certain values, if the statistic is unbiased (centered at the parameter it is meant to estimate), and how much variability the statistic has about its mean.

Goal: Estimate the proportion, p, of a population that belongs to a particular category (i.e. find the proportion of Americans that “approve” of the job GW is doing as President). Take a random sample of Americans, of size n and count the number of Americans in the sample that “approve.” Suppose we poll 110 Americans and 45 “approve.” Notation: X: The number (or count) of items in sample that are in the category. n: The sample size : The sample proportion (i.e. the proportion of the sample in the category) What are the values of X, n, and for this example? Section 5.1: Sampling Distributions for Counts and Proportions

Binomial Distributions for Sample Counts - X (page 335) Consider Example 5.2…. Does the example in the previous slide constitute a binomial setting? If so, identify, what is a “success,” n, and p for the example…. See technical note on the middle of page 337! Some Common Binomial Settings: Coins, Dice, etc.

Finding Probabilities for Binomially Distributed Counts Let’s work a few examples (coin, dice, etc.)! Calculator Commands: binompdf(n, p, value) gives probability (likelyhood) that X=value. binomcdf(n, p,value) gives probability that X is less than or equal to the value, i.e. that X=0 or X=1 or …X=value.

For the scenario in Problem 5.6, how likely is it that your sample will contain: (a) Exactly 4 “successes.” Hint: binompdf(n,p,value) (b) At most 4 “successes.” Hint: binomcdf(n,p,value) (c) Between 4 and 8, inclusive, “successes.” (d) On average, how many “successes” will your sample have? (see next slide!) Finding Probabilities for Sample Counts

The Binomial Mean and Standard Deviation Answer to Part (d) on Previous Slide?

Finding Probabilities for Sample Proportions Example 5.8 – Converting a probability about a sample proportion to a probability about a sample count. Note that the sample proportion is not binomial since it is not a count! How is the sample proportion distributed? What are the values defined above for this example?

Notice the shape, center, and variability of the sampling distribution of for n=2500.

Let’s Use Two Cool Applets to Visualize Each of These: Sampling Distribution of X: Applet on CD (CLT Binomial) Sampling Distribution of : Sampling Sim

The Sampling Distribution of Let’s rework Example 5.8 using the Normal Approximation! (Example 5.10 page 345, see illustration, next slide…)

Sampling Distribution of Normal Approximation of Sampling Distribution of

Section 5.2: The Sampling Distribution of a Sample Mean Population (Individuals) Sampling Distribution Of Means (Averages) for n=80 Averages are less variable than individual observations. Averages are more normal than individual observations. The Big Ideas:

Properties of the Sampling Distribution of a Sample Mean

Remember: IF the population is normal then the sampling distribution of the sample mean (for fixed n is normal) otherwise, via the CLT the sampling distribution of the sample mean becomes approximately normal as n increases! How fast does the CLT work? Let’s check it out via Sampling Sim!

The CLT In Action! Individual Measurements (Population Distribution) with Population Mean of 1 Averages (n=2) (Sampling Distribution of Sample Mean when n=2) Averages (n=10) (Sampling Distribution of Sample Mean when n=10) Averages (n=25) (Sampling Distribution of Sample Mean when n=25)

Let’s Work Some Problems! Problem 5.34 (page 370) Problem 5.40 (page 371)