This Discussion Problem comes from Chapter 7: Confidence Intervals in Introduction To Statistics: Think & Do by Scott Stevens. Please show work to any equations that are done. Thank you!

Chapter 7: Discussion 1 117 Chapter 7: Discussion 1 Nate Silver - 2012 Election Nathaniel (Nate) Silver is a popular statistician and writer. He first gained recognition for developing a system to forecast the performance of Major League Baseball players. More recently he gained further accolades by correctly predicting the winner of the 2012 presidential elections in all 50 states. The methods he used were more numerous and sophisticated than this demonstration can present. However, this demonstrative example gives some insight as to how he was able to predict a winner when everyone else was saying Too Close to Call. The data in this demonstration is contrived though it represents the idea behind a relatively simple concept called aggregate polling. 1. Individual Polls: Suppose there were three separate pre-election polls by three different polling firms in a given swing state. Pollster 1 Confidence Intervals n = 1600 P 99% interval 95% interval 90% interval 80% inteval Obama 49 (0.458, 0.522 (0.466, 0.514) (0.469, 0.511) (0.474, 0.506) Romney .45 (0.418, 0.482) (0.426, 0.474) (0.430, 0.470) (0.434, 0.466) Pollster 2 Confidence Intervals n = 600 p 99% interval 95% interval 90% interval 80% inteval Obama 47 (0.418, 0.522) (0.430, 0.510) (0.436, 0.504) (0.444, 0.496) Romney .48 (0.427, 0.533) (0.440, 0.520) (0.446, 0.514) (0.454, 0.506) Pollster 3 Confidence Intervals n = 2800 p 99% interval 95% interval 90% interval 80% inteval Obama .48 (0.456, 0.504) (0.461, 0.499) (0.464, 0.496) (0.468, 0.492 Romney 45 (0.426, 0.474) (0.432, 0.468) (0.435, 0.465) (0.438, 0.462) Looking at each of these individually, why would it be difficult to predict a winner? 2. Aggregate Poll: Assuming each poll was conducted properly (strict adherence to simple random sampling) you can aggregate the polls into one. This is done by getting a weighted average of the individual proportions where the weights are the sample sizes. Now you have an aggregate poll where n is a lot bigger. Aggregate Poll Confidence Intervals n = 5000 p 99% interval 95% interval 90% interval 80% inteval Obama 48 (0.464, 0.500) (0.468, 0.496) (0.470, 0.494) (0.473, 0.491) Romney 45 (0.435, 0.472) (0.440, 0.467) (0.442, 0.465) (0.445, 0.463) Looking at the aggregate confidence intervals, is it easier to predict a winner? Who would you pick and how confident are you