Chapter 19 Chapter19 Presentation 1213 Confidence Intervals for Proportions Copyright 2009 Pearson Education Inc 1 The Margin of Error When asked When you see or hear about an athlete breaking a world record in Swimming are you suspicious that the athlete used performance enhancing drugs 24 of 1007 US adults surveyed responded yes with a margin or error of 3 percentage points What does a margin of error mean what does it tell us and where does it come from Chapter19 Presentation 1213 Copyright 2009 Pearson Education Inc 2 The Margin of Error Cont Polling agencies have agreed what to report as the Margin of Error If the poll like the one on the previous slide is designed to estimate a population proportion the Margin of Error is approximately two times SD p Chapter19 Presentation 1213 Copyright 2009 Pearson Education Inc pq n 3 But what if we don t know p The formula on the previous slide is the standard deviation of the sampling distribution of the sample proportion you learned in Chapter 18 That formula involves the population proportion p Usually p is what we d like to know and why we took a sample to begin with Chapter19 Presentation 1213 Copyright 2009 Pearson Education Inc 4 But what if we don t know p Cont If we don t know p we re stuck right Not really We will use our sample statistic estimate the population parameter p in that formula to Whenever we estimate the standard deviation of a sampling distribution we call it a standard error Chapter19 Presentation 1213 Copyright 2009 Pearson Education Inc 5 Standard Error For a sample proportion the standard error is SE p pq n We can use SE p in place of SD p as an estimate when describing the sampling distribution Chapter19 Presentation 1213 Copyright 2009 Pearson Education Inc 6 Analyzing the World Record Swimmers Poll 24 responded they were suspicious of performance enhancing drug use among swimmers who break world records Is the proportion of all US adults who are suspicious close to 24 Probably But how close We will build an interval around 24 that we are pretty confident contains the true population proportion Chapter19 Presentation 1213 Copyright 2009 Pearson Education Inc 7 A Confidence Interval By the 68 95 99 7 Rule we know about 68 of all possible samples will have p s within 1 SE of p about 95 of all possible samples will have p s within 2 SEs of p about 99 7 of all possible samples will have p s within 3 SEs of p We can look at this from p s point of view Chapter19 Presentation 1213 Copyright 2009 Pearson Education Inc 8 A Confidence Interval cont Consider the 95 level There s a 95 chance that is no more than 2 SEs away from p Thus we re 95 confident that p is no more than 2 SEs away from So if we reach out 2 SEs we are 95 sure that p will be in that interval In other words if we reach out 2 SEs in either direction of we can be 95 confident that this interval contains the true proportion p This is called a 95 confidence interval Chapter19 Presentation 1213 Copyright 2009 Pearson Education Inc 9 A Confidence Interval cont Chapter19 Presentation 1213 Copyright 2009 Pearson Education Inc 10 What Does 95 Confidence Really Mean cont The figure to the right shows that some of our confidence intervals from 20 random samples capture the true proportion the green horizontal line while others do not Chapter19 Presentation 1213 Copyright 2009 Pearson Education Inc 11 What Does 95 Confidence Really Mean cont Our confidence is in the process of constructing the interval not in any one interval itself Thus we expect 95 of all 95 confidence intervals to contain the true parameter that they are estimating Chapter19 Presentation 1213 Copyright 2009 Pearson Education Inc 12 75 Free Throw Shooter Simulation Revisited The File Ch19 Free Throws Confidence Intervals contains several sections results for our in class simulation Recall each team simulated 50 free throws for a 75 free throw shooter and the number of made free throws were recorded The number of made free throws can be divided by 50 to obtain the proportion of made free throws 330 different teams results are represented in the file Chapter19 Presentation 1213 Copyright 2009 Pearson Education Inc 13 75 Free Throw Shooter Simulation Revisited The 330 values in the column Sample Proportion can be thought of as 330 p values 95 confidence intervals were constructed for each of the 330 proportions see the formulas that make up some of the columns What percent of the 330 confidence intervals captured the true proportion of p 0 75 Chapter19 Presentation 1213 Copyright 2009 Pearson Education Inc 14 Analyzing the World Record Swimmers Poll cont 24 responded they were suspicious of performance enhancing drug use among swimmers who break world records The standard error is SE p p q n A 95 confidence interval is Chapter19 Presentation 1213 Copyright 2009 Pearson Education Inc 15 Analyzing the World Record Swimmers Poll cont Our 95 confidence interval for p is 213 267 Our interval estimate has a margin of error of which is 0 027 or 2 7 percentage points We re 95 confident that p is within the above interval So does this interval contain p Chapter19 Presentation 1213 Copyright 2009 Pearson Education Inc 16 Margin of Error Certainty vs Precision The extent of the interval on either side of p is called the margin of error ME In general confidence intervals have the form estimate ME The more confident we want to be given the same sample data the larger our ME needs to be makes the interval wider Chapter19 Presentation 1213 Copyright 2009 Pearson Education Inc 17 Margin of Error Certainty vs Precision cont Chapter19 Presentation 1213 Copyright 2009 Pearson Education Inc 18 Margin of Error Certainty vs Precision cont To be more confident we wind up being less precise We need more values in our confidence interval to be more certain Because of this every confidence interval is a balance between certainty and precision Chapter19 Presentation 1213 Copyright 2009 Pearson Education Inc 19 Margin of Error Certainty vs Precision cont The choice of confidence level is somewhat arbitrary The most commonly chosen confidence levels are 90 95 and 99 but any percentage can be used Chapter19 Presentation 1213 Copyright 2009 Pearson Education Inc 20 Critical Values The 2 in p 2SE p our 95 confidence interval came from the 68 95 99 7 Rule Using technology we find that a more exact value for our 95 confidence interval is 1 96 instead of 2 We call 1 96 the critical value and denote it z Chapter19 Presentation
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