Virginia Mathematics Teacher Spring 2017

the proportion p. This means that the actual interval for the proportion, p, of people expected to vote for each candidate most likely lies within this range. The confidence interval also has a confidence level associated with it. The confidence level is the probability stated as a percentage that the confidence interval contains the population parameter when the sampling is repeated a very large number of times. Typically, pollsters use a 95% confidence level. This means, 95 out of 100 times the confidence intervals calculated from 100 samples would contain the true proportion, p. Next, these polling results are examined more closely to show how the confidence interval sheds light on the meaning of the numbers. The polling results show that 49% of the people polled claimed they would vote for Clinton with a 3.5% margin of error. This means the range of the proportion of voters who claim they will vote for Clinton is from 45.5% to 52.5 %. This suggests that Clinton would receive between 45.5% and 52.5% of the popular vote. Likewise, the polling results show that 40.5% of the people polled said they would vote for Trump with a 3.5% margin of error. This means, that it is expected that Trump would receive from 40.5% to 47.5% of the popular vote. Comparing these two intervals on a number line, we observe an overlap (see Figure 1). The overlap is approximately 1/3 of each candidates’ confidence interval. The confidence intervals state that if the sampling were repeated 100 times we would expect

that the true proportion of the popular vote for each candidate would lie within their confidence interval almost all of the time. What is unknown is the true proportion of voters within these intervals. Every poll the first author examined in the month leading up to the election showed Clinton leading the race. Yet, understanding the statistics behind the polling numbers suggests that neither candidate is clearly ahead. Other polling organizations perform a meta- analysis to identify the proportion of popular vote for the candidates. A meta-analysis combines data from multiple polling sources. The reason to combine data is to reduce the margin of error, while maintaining the same level of confidence, 95%. The reduced margin of error creates a narrower confidence interval, which generates results with greater precision. For example, the organization, Real Clear Politics, performed a meta -analysis to determine the proportion of voters for each candidate. The margin of error, E, is smaller, 1.9% compared to the earlier polling organization that reported a margin of error, E, of 3.5%. The Real Clear Politics organization polling results are shown below. Clinton:

48.2% 1.9% 46.3% 50.1% p E to r r

Figure 1. Comparing two intervals with an overlap

Figure 2 . Comparing two intervals with a larger overlap

Virginia Mathematics Teacher vol. 43, no. 2

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