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UT Knoxville STAT 201 - Chapter 19

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1Chapter19 Presentation 1213Copyright © 2009 Pearson Education, Inc.Chapter 19 Confidence Intervals for ProportionsChapter19 Presentation 1213Copyright © 2009 Pearson Education, Inc.2The 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 1213Copyright © 2009 Pearson Education, Inc.3 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:SDˆppqnThe Margin of Error (Cont.)Chapter19 Presentation 1213Copyright © 2009 Pearson Education, Inc.4But 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 1213Copyright © 2009 Pearson Education, Inc.5But 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 to estimate the population parameter p in that formula. Whenever we estimate the standard deviation of a sampling distribution, we call it a standard error.Chapter19 Presentation 1213Copyright © 2009 Pearson Education, Inc.6 For a sample proportion, the standard error is We can use in place of as an estimate when describing the sampling distribution. Standard ErrorEˆˆˆSppqnSEˆpSDˆpChapter19 Presentation 1213Copyright © 2009 Pearson Education, Inc.7Analyzing 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 1213Copyright © 2009 Pearson Education, Inc.8A Confidence Interval By the 68-95-99.7% Rule, we know about 68% of all possible samples will have ’s within 1 SE of p about 95% of all possible samples will have ’s within 2 SEs of p about 99.7% of all possible samples will have ’s within 3 SEs of p We can look at this from ’s point of view…ˆpˆpˆpˆpChapter19 Presentation 1213Copyright © 2009 Pearson Education, Inc.9A 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 1213Copyright © 2009 Pearson Education, Inc.10A Confidence Interval (cont.)Chapter19 Presentation 1213Copyright © 2009 Pearson Education, Inc.11What 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 1213Copyright © 2009 Pearson Education, Inc.12What 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 1213Copyright © 2009 Pearson Education, Inc.1375% 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 1213Copyright © 2009 Pearson Education, Inc.1475% 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 1213Copyright © 2009 Pearson Education, Inc.15Analyzing the World Record Swimmers Poll (cont.)SEˆpˆpˆqn 24% responded they were suspicious of performance enhancing drug use among swimmers who break world records.  The standard error is: A 95% confidence interval isChapter19 Presentation 1213Copyright © 2009 Pearson Education, Inc.16 Our 95% confidence interval for p is (.213, .267)  Our interval estimate has a margin of error ofwhich 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? Analyzing the World Record Swimmers Poll (cont.)Chapter19 Presentation 1213Copyright © 2009 Pearson Education, Inc.17Margin of Error: Certainty vs. Precision The extent of the interval on either side of 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).ˆpChapter19 Presentation 1213Copyright © 2009 Pearson Education, Inc.18Margin of Error: Certainty vs. Precision (cont.)Chapter19 Presentation 1213Copyright © 2009 Pearson Education, Inc.19Margin 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


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UT Knoxville STAT 201 - Chapter 19

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