Chapter 18 Sampling Distribution Models Note A few concepts from Chapter 16 are contained within these slides Chapter18 Presentation 1213 Copyright 2009 Pearson Education Inc 1 Random 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 1213 Copyright 2009 Pearson Education Inc 2 Discrete 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 1213 Copyright 2009 Pearson Education Inc 3 Discrete 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 1213 Copyright 2009 Pearson Education Inc 4 Probability 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 1213 Copyright 2009 Pearson Education Inc 5 Probability 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 p x 1 Chapter18 Presentation 1213 Copyright 2009 Pearson Education Inc 6 Example Flip a fair coin 5 times X the number of heads x p X x 0 1 2 3 4 5 03 16 31 31 16 03 Is this a valid pmf Chapter18 Presentation 1213 Copyright 2009 Pearson Education Inc 7 Back 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 1213 Copyright 2009 Pearson Education Inc 8 What is Expected Of particular interest is the average value we expect a random variable to take on in the long run notated population mean or E X for expected value E X average value of the random variable in the long run very long run We are also interested in the standard deviation which is the typical difference between the actual values of the random variable and the average value Chapter18 Presentation 1213 Copyright 2009 Pearson Education Inc 9 Can 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 1213 Copyright 2009 Pearson Education Inc 10 Where 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 1213 Copyright 2009 Pearson Education Inc 11 Introduction 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 808 U 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 1213 Copyright 2009 Pearson Education Inc 12 Review 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 y Chapter18 Presentation 1213 Copyright 2009 Pearson Education Inc 13 Proportions 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 1213 Copyright 2009 Pearson Education Inc 14 Proportions Vary from Sample to Sample Cont The following are the results from several sections combined Chapter18 Presentation 1213 Copyright 2009 Pearson Education Inc 15 Definition 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 Sample Values Sample proportion of odd numbers Sampling Distribution Sample Frequency Proportion 0 0 0 0 2 0 0 4 6 0 6 12 0 8 3 1 0 0 Population proportion p 4 7 57 1 Average of all 21 sample proportions 12 0 21 57 1 Chapter18 Presentation 1213 Sum 12 0 Copyright 2009 Pearson Education Inc 16 Properties of Sampling Distribution 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 Chapter18 Presentation 1213 Copyright 2009 Pearson Education Inc 17 Properties of Sampling Distribution Cont Expected Value of The sample proportions get it right on average the mean of all possible s is equal to p the expected value of is p the true population proportion Standard Deviation of The typical difference between and p i e the difference between what we measure and the true population proportion is known as the standard deviation of Chapter18 Presentation 1213 Copyright 2009 Pearson Education Inc 18 Standard deviation of Amazingly there is a formula for the SD of given the population proportion p q 1 p and the sample size n SD pq n Interpretation the
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