1Chapter18 Presentation 1213Copyright © 2009 Pearson Education, Inc.Chapter 18 Sampling Distribution Models Note: A few concepts from Chapter 16 are contained within these slides.Chapter18 Presentation 1213Copyright © 2009 Pearson Education, Inc.2Random Variables A random variable assumes a value based on the outcome of a random event. Random variables are denoted by a capital letter such as X.Chapter18 Presentation 1213Copyright © 2009 Pearson Education, Inc.3Discrete Random Variables A discrete random variable is a variable that can take on only whole numbers. We denote a particular value that a discrete random variable can take on with a lower-case letter such as x.Chapter18 Presentation 1213Copyright © 2009 Pearson Education, Inc.4Discrete Random Variables (cont’d) Examples: X = the # of students that come to class x = 27,94,54,… X = the # of people, out of 10, that believe in ghosts x = 1, 7, 5,… X = the # of subs sold daily from Subway x = X = the # of goals scored in a MLS game x =Chapter18 Presentation 1213Copyright © 2009 Pearson Education, Inc.5Probability Distribution A probability distribution for a random variable consists of: The collection of all possible values of a random variable, and the probabilities that the values occur.Chapter18 Presentation 1213Copyright © 2009 Pearson Education, Inc.6Probability Distribution (Cont’d) For a discrete random variable, the probability distribution of outcomes is called a probability mass function (pmf) and is represented with p(x). A valid pmf must have these characteristics: p(x) ≥ 0 for all x Σp(x) = 1Chapter18 Presentation 1213Copyright © 2009 Pearson Education, Inc.7Example Flip a fair coin 5 times. X = the number of heads Is this a valid pmf?x012345p(X=x) .03 .16 .31 .31 .16 .03Chapter18 Presentation 1213Copyright © 2009 Pearson Education, Inc.8Back to the Discrete Random Variables Examples – What is expected? X = the # of students that come to class X = the # of people, out of 10, that believe in ghosts X = the # of subs sold daily from Subway X = the # of goals scored in a MLS game What value do you expect for each of these examples, in the long run, on average?Chapter18 Presentation 1213Copyright © 2009 Pearson Education, Inc.9What is Expected? Of particular interest is the average value we expect a random variable to take on in the long run, notated μ (population mean) or E(X)for “expected value”. μ = E(X) = average value of the random variable, in the long run (very long run) We are also interested in the standard deviation σ which is the typical difference between the actual values of the random variable and the average value.Chapter18 Presentation 1213Copyright © 2009 Pearson Education, Inc.10Can the Expected Value and Standard Deviation be Calculated? If one knew the probability mass function of a discrete random variable, the expected value and standard deviation of x could be computed. In most phenomenon we don’t fully understand (but want to understand), the probability mass function is not known.Chapter18 Presentation 1213Copyright © 2009 Pearson Education, Inc.11Where Is This Going? In the material that follows, we will assume that we do know the probability distribution of a particular discrete random variable (and so, we will be able to calculate the expected value and standard deviation). We will then use this expected value and standard deviation to help us define what are “unusual” or “rare” outcomes.Chapter18 Presentation 1213Copyright © 2009 Pearson Education, Inc.12Introduction to Sampling Distributions In Nov. 2005, Harris Poll asked 889 U.S. adults “Do you believe in ghosts?” 40% of the respondents said they did. At about the same time, CBS News poled 808U.S. adults and asked the same question. 48% of the respondents said they did. Why are these results different? Is this surprising? Which one is right? How reliable are sample proportions?Chapter18 Presentation 1213Copyright © 2009 Pearson Education, Inc.13Review of Some Terminology Population – the set of all objects or individuals we wished we had data on (e.g., all U.S. adults). Sample – the set of objects that we actually have data on (e.g., a survey of 889 U.S. adults). We wished we knew the population proportion p, the fraction or percent of U.S. adults who believe in ghosts, but we only have the sample proportion(pronounced “p hat”). This chapter will talk about the relationship between p and , as well as between y and .yChapter18 Presentation 1213Copyright © 2009 Pearson Education, Inc.14Proportions Vary from Sample to Sample For example, in the in-class simulation, teams reported the number of made free throws out of 50 attempts for a fictitious 75% free throw shooter. We can divide each team’s result by 50 and obtain the proportion of made free throws. You can think of each team’s proportion of free throws made as a sample proportion (p) from a population with p=.75.Chapter18 Presentation 1213Copyright © 2009 Pearson Education, Inc.15Proportions Vary from Sample to Sample (Cont.) The following are the results from several sections combined:Chapter18 Presentation 1213Copyright © 2009 Pearson Education, Inc.16Definition of a “Sampling Distribution” Example: imagine the population has values 1, 2, 4, 5, 7, 8, 9. We are interested in the proportion of odd numbers in a sample of size 5. We can enumerate all possible samples that can be taken from the population, and can tabulate the proportion of odd values for each sample. The distribution of all possible sample proportions is known as the sampling distribution of the sample proportion.Sample ProportionFrequency0.0 00.2 00.4 60.6 120.8 31.0 0Population proportion p = 4 / 7 = 57.1%Average of all 21 sample proportions = 12.0/21 = 57.1%Sampling DistributionSample #Sample ValuesSample proportion of odd numbersSum = 12.0Chapter18 Presentation 1213Copyright © 2009 Pearson Education, Inc.17 The sample proportion will differ from sample to sample. The distribution of all possible sample proportions is called the sampling distribution of the sample proportion.Properties of Sampling DistributionChapter18 Presentation 1213Copyright © 2009 Pearson Education, Inc.18 Expected Value of : The sample proportions get it right on average (the mean
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