/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 17 The following table shows the re... [FREE SOLUTION] | 91Ó°ÊÓ

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The following table shows the result of the 2012 presidential election along with the vote predicted by several organizations in the days before the election. The sample sizes were typically about 1000 to 2000 people. The percentages for each poll do not sum to 100 because of voters who indicated they were undecided or preferred another candidate. a. Treating the sample sizes as 1000 each, find the approximate margin of error. b. Do most of the predictions fall within the margin of error of the actual vote percentages? Considering the relative sizes of the sample, the population, and the undecided factor, would you say that these polls had good accuracy?

Short Answer

Expert verified
The margin of error is approximately 3.1%. Most predictions likely fall within this range, suggesting reasonable accuracy.

Step by step solution

01

Understanding Margin of Error

The margin of error is a statistic expressing the amount of random sampling error in a survey's results. It is often calculated as \( Z \times \sqrt{\frac{p(1-p)}{n}} \), where \( p \) is the observed proportion, \( n \) is the sample size, and \( Z \) is the Z-score for the desired confidence level (typically 1.96 for 95% confidence).
02

Estimating Proportion

Since we don't have a given proportion for each candidate, we often use \( p = 0.5 \) for a maximum margin of error. This simplifies the calculation assuming the worst-case scenario.
03

Calculating Margin of Error

Using \( p = 0.5 \), the margin of error for each poll with a sample size of 1000 at 95% confidence level is computed as follows:\[ \text{Margin of Error} = 1.96 \times \sqrt{\frac{0.5 \times (1 - 0.5)}{1000}} \approx 0.031 \text{ or } 3.1\% \].
04

Comparing Predictions to Actual Results

Examine whether the predicted vote percentages fall within the calculated 3.1% margin of error of the actual election results. Subtract the actual vote percentages from the predictions and check if the difference is less than the margin of error.
05

Assessing Poll Accuracy

To determine if the polls had good accuracy, consider if most of the predictions fell within the margin of error and note influences like sample size and undecided voters. If a majority of predictions are within the margin, the polls might be seen as reasonably accurate even with undecided voters impacting the outcome.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Presidential Election Polls
Presidential election polls are a common way to gauge public opinion before an election occurs. These polls are conducted by sampling a small group of people from a larger population to predict how the entire population might vote. The outcome of these polls can be crucial as they help candidates, strategists, and political analysts understand potential election results and shape political strategies accordingly.

Polls serve multiple functions:
  • They provide a snapshot of public opinion at a given time.
  • They help identify trends in voter preferences.
  • They allow campaigns to adjust their messages based on voter feedback.

The effectiveness of these polls, however, depends on various factors. Ensuring a representative sample, accurate question phrasing, and minimizing bias are all essential to obtain reliable and valid results. Nonetheless, even with meticulous planning, polls can sometimes deviate from actual election outcomes due to the inherent unpredictability of voting behavior and last-minute changes in voter decisions.
Statistical Accuracy
Statistical accuracy in the context of presidential election polls means how closely the poll results match the actual election outcomes. Two key components that affect statistical accuracy are the margin of error and the confidence level.

  • Margin of Error: This is an indicator of the expected range within which the true population parameter lies. It accounts for variability due to random sampling and is a function of sample size and assumed population proportion.

  • Confidence Level: Usually set at 95%, this indicates the likelihood that the true population parameter falls within the calculated margin of error. It is often denoted by a Z-score of 1.96.

A high statistical accuracy means most poll predictions should fall within their respective margins of error. For instance, if a poll predicts that a candidate will receive 48% of the vote with a 3% margin of error, there is a high probability that the actual vote will range from 45% to 51% if the poll is statistically accurate.

Polls with high accuracy can be trusted as reliable indicators of election outcomes, assuming other conditions such as sample representation and survey methodology are sound.
Sample Size Effect
The size of the sample taken in a survey directly impacts the margin of error and, by extension, the reliability of poll results. In presidential election polls, larger sample sizes generally lead to smaller margins of error, providing a more accurate prediction of the population's preferences.

There are key reasons why sample size matters:
  • Reduced Variability: Larger samples tend to reduce the sampling error, thus tightening the margin of error.
  • Representation: A bigger sample is more likely to represent the diverse opinions of the entire population accurately.
  • Confidence in Results: More rounded samples often lead to greater confidence levels, thereby enhancing the credibility of the polls.

For example, when a sample size increases from 1000 to 2000 people, the margin of error will decrease, improving the poll's predictive strength. However, practical considerations such as time, cost, and logistical challenges need to be balanced against the ideal of increasing sample sizes indefinitely.

The balance between sample size and available resources is crucial for conducting reliable and effective presidential election polls.

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Most popular questions from this chapter

Consider the population of all undergraduate students at your university. A certain proportion have an interest in graduate studies. Your friend randomly samples 40 undergraduates and uses the sample proportion of those who have an interest in graduate studies to predict the population proportion at the university. You conduct an independent study of a random sample of 40 undergraduates and find the sample proportion of those are interested in graduate studies. For these two statistical studies, a. Are the populations the same? b. How likely is it that the samples are the same? Explain. c. How likely is it that the sample proportions are equal? Explain.

An analysis by Professor Peter M Rothwell and his colleagues (Nuffield Department of Clinical Neuroscience, University of Oxford, UK) published in 2012 in the medical journal The Lancet (http://www. thelancet.com) assessed the effects of daily aspirin intake on cancer mortality. They looked at individual patient data from 51 randomized trials \((77,000\) participants) of daily intake of aspirin versus no aspirin or other anti-platelet agents. According to the authors, aspirin reduced the incidence of cancer, with maximum benefit seen when the scheduled duration of trial treatment was five years or more and resulted in a relative reduction in cancer deaths of about \(15 \%\) (562 cancer deaths in the aspirin group versus 664 cancer deaths in the Control group). Specify the aspect of this study that pertains to (a) design, (b) description, and (c) inference.

Euthanasia The General Social Survey asked, in \(2012,\) whether you would commit suicide if you had an incurable disease. Of the 3112 people who had an opinion about this, \(1862,\) or \(59.8 \%,\) would commit suicide. a. Describe the population of interest. b. Explain how the sample data are summarized using descriptive statistics. c. For what population parameter might we want to make an inference?

A poll was conducted by Ipsos Public Affairs for Global News. It was conducted between May 18 and May \(20,2016,\) with a sample of 1005 Canadians. It was found \(73 \%\) of the respondents "agree" that the Liberals should not make any changes to the country's voting system without a national referendum first. (http://globalnews.ca/news/). Find the approximate margin of error if the poll had been based on a sample of size (a) \(n=900,\) (b) \(n=1600,\) and \((\mathrm{c}) n=2500 .\) Explain how the margin of error changes as \(n\) increases.

We'll see that the amount by which statistics vary from sample to sample always depends on the sample size. This important fact can be illustrated by thinking about what would happen in repeated flips of a fair coin. a. Which case would you find more surprising - flipping the coin five times and observing all heads or flipping the coin 500 times and observing all heads? b. Imagine flipping the coin 500 times, recording the proportion of heads observed, and repeating this experiment many times to get an idea of how much the proportion tends to vary from one sequence to another. Different sequences of 500 flips tend to result in proportions of heads observed which are less variable than the proportion of heads observed in sequences of only five flips each. Using part a, explain why you would expect this to be true.

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