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Chapter 14: Problem 19

Refer to the following story: The city of Cleansburg has 8325 registered voters. There is an election for mayor of Cleansburg, and there are three candidates for the position: Smith, Jones, and Brown. The day before the election a telephone poll of 680 randomly chosen registered voters produced the following results: 306 people surveyed indicated that they would vote for Smith, 272 indicated that they would vote for Jones and 102 indicated that they would vote for Brown. Given that in the actual election Smith received \(42 \%\) of the vote, Jones \(43 \%\) of the vote, and Brown \(15 \%\) of the vote, find the sampling errors in the survey expressed as percentages.

Short Answer

Expert verified
The sampling errors expressed in percentages are -3% for Smith, 3% for Jones, and 0% for Brown. This indicates that the survey overestimated Smith's support by 3%, underestimated Jones's support by 3%, and accurately predicted Brown's support.

Step by step solution

01

Calculate Survey Percentages

First, calculate what percent of those polled planned to vote for each candidate: Smith: \(\frac{306}{680} \times 100 = 45 \% \)Jones: \(\frac{272}{680} \times 100 = 40 \% \)Brown: \(\frac{102}{680} \times 100 = 15 \% \)
02

Calculate Real Percentages

Next, record the actual election results, which are already provided as percentages:Smith: \(42 \% \)Jones: \(43 \% \)Brown: \(15 \% \)
03

Calculate Sampling Error

Subtract the survey percentage (Step 1) from the election percentage (Step 2) for each candidate to find the sampling error:Smith: \( 42 \% - 45 \% = -3 \% \)Jones: \( 43 \% - 40 \% = 3 \% \)Brown: \( 15 \% - 15 \% = 0 \% \)

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Key Concepts

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

Election Polling
An election polling is a tool used to gauge which candidate is likely to win an election. In Cleansburg's mayoral race, a poll surveyed 680 out of the 8325 registered voters. It involved asking people who they intend to vote for to predict the election's outcome.
Pollsters choose a smaller group of voters, known as a sample, to infer the preferences of the entire voter population. This is usually less costly and time-consuming than surveying everyone. However, it's important to remember that polls are predictions and not always exact reflections of election outcomes.
  • Polls provide a snapshot of voter intentions at a specific time.
  • Sampling methods should be random to avoid biased results.
  • Polling ahead of an election can often influence voter perceptions and behaviors.
Understanding election polls' limitations is essential to appropriately interpreting their results.
Percentage Calculation
Percentage calculation is critical in understanding how many people in a poll plan to vote for each candidate, and this is done by comparing surveyed numbers to the total sample.
For example, in Cleansburg, Smith received support from 306 voters in a survey of 680. To find his percentage, divide 306 by 680, and then multiply by 100: Smith: \(\frac{306}{680} \times 100 = 45\%\) Perform the same calculations for the other candidates to understand their support.
  • Jones: \(\frac{272}{680} \times 100 = 40\%\)
  • Brown: \(\frac{102}{680} \times 100 = 15\%\)
Calculating percentages this way helps us understand the distribution of support among candidates. It's a straightforward step in analyzing poll data and makes comparisons easier.
Survey Analysis
Survey analysis involves comparing predicted outcomes from a poll with actual election results. This is where understanding sampling error becomes crucial. Sampling error is the difference between the poll results and actual election results. In this context, it's useful to identify how accurate a poll was in predicting the election outcome.

For Cleansburg's election, let's analyze the survey errors:
  • For Smith, the survey showed 45% while he got 42% in real elections, indicating a sampling error of \(-3\%\).
  • Jones's error was \(3\%\) since he had 43% in real elections but was predicted at 40% in polls.
  • For Brown, there was no error as both the survey and real elections showed the same percentage, \(15\%\).
These differences underscore the necessity of interpreting survey data cautiously. Noticing discrepancies aids in refining poll methodologies and understanding voter behavior.

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Most popular questions from this chapter

A big concert was held at the Bowl. Men and women had to go through separate lines to get into the concert (the women had to have their purses checked). Once everyone was inside, total attendance at the concert had to be recorded. The turnstile counters on the female entrance showed a total count of 1542 females, but the turnstile counters on the male entrance were broken and there was no exact record of how many males attended. A sample taken from the 200 seats in Section A showed 121 females and 79 males in that section. Using the numbers from Section A, estimate the total attendance at the concert. (Hint: The proportion of females at the concert should be roughly the same as the proportion of females in Section A.)

The critically endangered Maui's dolphin is currently restricted to a relatively small stretch of coastline along the west coast of New Zealand's North Island. The dolphins are "captured" by just collecting samples of DNA and "tagged" by identifying their DNA fingerprint. A \(2010-2011\) capturerecapture study "captured" and "tagged" 26 Maui's dolphins in 2010. In 2011, 27 Maui's dolphins were "recaptured" and through their DNA, 12 were identified as having been "tagged" in 2010. Based on these figures, estimate the population of Maui's dolphins in 2011. [Source: Oremus, M., et al, "Distribution, group characteristics and movements of the critically endangered Maui's Dolphin (Cephalorhynchus hectori maui)." Endangered Species Research, preprint.]

To estimate the population in a rookery, 4965 fur seal pups were captured and tagged in early August. In late August, 900 fur seal pups were captured. Of these, 218 had been tagged. Based on these figures, estimate the population of fur seal pups in the rookery. [Source: Chapman and Johnson, "Estimation of Fur Seal Pup Populations by Randomized Sampling," Transactions of the American Fisheries Society, 97 (July 1968), 264-270.

Refer to the following story: The city of Cleansburg has 8325 registered voters. There is an election for mayor of Cleansburg, and there are three candidates for the position: Smith, Jones, and Brown. The day before the election a telephone poll of 680 randomly chosen registered voters produced the following results: 306 people surveyed indicated that they would vote for Smith, 272 indicated that they would vote for Jones and 102 indicated that they would vote for Brown. (a) Give the sampling proportion for this survey. (b) Give the sample statistic estimating the percentage of the vote going to Smith.

Refer to the following story (see also Exercise 32): The Dean of Students at Tasmania State University wants to determine how many undergraduates at TSU are familiar with a new financial aid program offered by the university. There are 15,000 undergraduates at \(T S U,\) so it is too expensive to conduct a census. The following sampling method, known as systematic sampling, is used to choose a representative sample of undergraduates to poll. Start with the registrar's alphabetical listing containing the names of all undergraduates. Randomly pick a number between 1 and \(100,\) and count that far down the list. Take that name and every 100 th name after it. For example, if the random number chosen is \(73,\) then pick the \(73 \mathrm{rd}, 173 \mathrm{rd}, 273 \mathrm{rd},\) and so forth, names on the list. (a) Find the sampling proportion. (b) Suppose that the survey had a response rate of \(90 \%\). Find the size \(n\) of the sample.
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