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Chapter 7: Problem 95

In the primaries leading up to the 2016 presidential election, the Business Insider reported that Bernie Sanders and Hilary Clinton were in a "statistical tie" in the polls leading up to the Vermont primary. Clinton led Sanders \(43 \%\) to \(35 \%\) in the polls, with a margin of error of \(5.2 \%\). Explain what this means to someone who may be unfamiliar with margin of error and confidence intervals.

Short Answer

Expert verified
The margin of error is 5.2%, which means the support for each candidate could be 5.2% higher or lower than the survey results. For Clinton, her support could range from 37.8% to 48.2%, and for Sanders, it could range from 29.8% to 40.2%. As Clinton's and Sanders' ranges overlap, they are in a 'statistical tie'. Even though Clinton has more support in the poll, it's not conclusive that she has more overall support, due to the margin of error. This range of potential support is the confidence interval, showing where we expect the true support to lie.

Step by step solution

01

Definition of Margin of Error

The margin of error gives an idea of how close the results of a survey can come to the true value. In this case, a margin of error of 5.2% suggests that the true proportion of voters who support each candidate could be 5.2% more or less than what the survey reports.
02

Application to the Poll Results

Applying the margin of error to the poll results, Clinton could actually have support from anywhere between 37.8% (43%-5.2%) to 48.2% (43%+5.2%). Similarly, Sanders could have support from anywhere between 29.8% (35%-5.2%) to 40.2% (35%+5.2%).
03

Understanding a Statistical Tie

A 'statistical tie' occurs when the ranges of support overlap considering the margin of error. In this case, because 37.8% to 48.2% (for Clinton) and 29.8% to 40.2% (for Sanders) have overlapping regions, despite Clinton leading in the poll, we cannot conclusively say she is leading due to the margin of error. Therefore, they are described as being in a 'statistical tie'.
04

Definition Of Confidence Intervals

Confidence intervals are related to the margin of error. The confidence interval is an estimate of where we expect the true population parameter to lie. In this case, it is reflected in the ranges we calculated in step 2.
05

Importance of Confidence Intervals and Margin of Error

The margin of error and the concept of confidence intervals are vital in understanding and interpreting survey or poll results. They provide a range within which the actual result may lie and indicate the level of precision of the survey/poll. Therefore, even if someone is leading in the polls, they may not actually be leading overall once you take the margin of error into account.

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Most popular questions from this chapter

A recent Monmouth University poll found that 675 out of 1008 randomly selected people in the United States felt that college and universities with big sports programs placed too much emphasis on athletics over academics. Assuming the conditions for using the CLT were met, use the Minitab output provided to answer these questions. $$\begin{aligned}&\text { Descriptive Statistics }\\\&\begin{array}{rrrr}\mathrm{N} & \text { Event } & \text { Sample p } & 95 \% \mathrm{Cl} \text { for } \mathrm{p} \\\\\hline 1008 & 675 & 0.669643 & (0.639648,0.698643)\end{array}\end{aligned}$$ a. Complete this sentence: I am \(95 \%\) confident that the population proportion believing that colleges and universities with big sports programs place too much emphasis on athletics over academics is between _____ and _______. Report each number as a percentage rounded to one decimal place. b. Suppose a sports blogger wrote an article claiming that the majority of Americans believe that colleges and university with big sports programs place too much emphasis on athletics over academics. Does this confidence interval support the blogger's claim? Explain your reasoning.

The city of Chicago provides an open data set of the number of WiFi sessions at all of its public libraries. For 2014 , there were an average of \(451,846.9\) WiFi sessions per month at all Chicago public libraries. What is the correct notation for the value \(451,846.9 ?\)

The website www.mlb.com compiles statistics on all professional baseball players. For the 2017 season, statistics were recorded for all 663 players. Of this population, the mean batting average was \(0.236\) with a standard deviation of \(0.064\). Would it be appropriate to use this data to construct a \(95 \%\) confidence interval for the mean batting average of professional baseball players for the 2017 season? If so, construct the interval. If not, explain why it would be inappropriate to do so.

In the 2018 study Closing the STEM Gap, researchers wanted to estimate the percentage of middle school girls who planned to major in a STEM field. a. If a \(95 \%\) confidence level is used, how many people should be included in the survey if the researchers wanted to have a margin of error of \(3 \%\) ? b. How could the researchers adjust their margin of error if they want to decrease the number of study participants?

A large collection of one-digit random numbers should have about \(50 \%\) odd and \(50 \%\) even digits, because five of the ten digits are odd \((1,3,5,7\), and 9\()\) and five are even \((0,2,4,6\), and 8\()\). a. Find the proportion of odd-numbered digits in the following lines from a random number table. Count carefully. $$\begin{array}{lll}57.283 \mathrm{pt} & 74834 & 81172 \\\\\hline 89281 & 48134 & 71185\end{array}$$ b. Does the proportion found in part a represent \(\hat{p}\) (the sample proportion) or \(p\) (the population proportion)? c. Find the error in this estimate, the difference between \(\hat{p}\) and \(p\) (or \(\hat{p}-p\) ).
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