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Chapter 9: Problem 27

The Gallup Organization conducts an annual survey on crime. It was reported that \(25 \%\) of all households experienced some sort of crime during the past year. This estimate was based on a sample of 1002 randomly selected adults. The report states, "One can say with \(95 \%\) confidence that the margin of sampling error is \(\pm 3\) percentage points." Explain how this statement can be justified.

Short Answer

Expert verified
The statement is justified because a 95% confidence level indicates that the results have not likely arisen by chance, given the sample size and the reported crime rate. The margin of error of 卤3% is within an acceptable range for a sample size of 1002, and it can be calculated using the formula for a binomial proportion.

Step by step solution

01

Understanding the Confidence Level

A 95% confidence level means that if the same survey were carried out 100 times, the results would fall within the given margin of error 95 times out of 100. This is a measure of the survey's reliability and the likelihood that its results have not arisen by chance.
02

Calculation of Margin of Error

The margin of error for a survey is a range within which the true value is likely to fall. It can be calculated for a binomial proportion using the formula \(ME = Z*鈭歱(1鈭抪)/n)\), where \(Z\) is the Z-critical value for the desired confidence level (1.96 for a 95% confidence level), \(p\) is the proportion observed (0.25 in this case), and \(n\) is the sample size (1002). This formula yields a margin of error of 卤3%.
03

Significance of Sample Size

The sample size is a crucial factor in determining the margin of error because a larger sample size generally yields more accurate estimates. Given a large enough sample size, like 1002 households in this case, a margin of error of 卤3% is justifiable.

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Key Concepts

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

Margin of Error
The margin of error is an essential concept in statistics which helps us understand the range within which the true value of a survey result is expected to fall. In the context of the Gallup Organization's survey, when they state that the margin of sampling error is 卤3 percentage points, they're saying that the actual percentage of households experiencing crime could realistically be 3 percentage points higher or lower than the 25% reported. Here's how it works:
  • It's calculated using this formula: \[ ME = Z \cdot \sqrt{\frac{p(1-p)}{n}} \]
  • For a 95% confidence level, the Z-critical value is 1.96.
  • In this survey, the observed proportion \(p\) is 0.25, and the sample size \(n\) is 1002.
  • Plug these values into the formula and you get a margin of error of about 卤3%.
This indicates a 95% certainty that the interval from 22% to 28% contains the true percentage of affected households. It's a crucial metric for understanding the precision and reliability of survey results.
Sample Size
Sample size refers to the number of observations or participants in a study or survey, and it plays a vital role in the accuracy of the study's results. A larger sample size generally provides more reliable estimates because it reduces the margin of error, making the confidence interval narrower. In the Gallup Organization's crime survey:
  • The sample size was 1002 households. This is relatively large, ensuring that their findings were more precise and could be generalized to the broader population.
  • With a large sample size, any variability or "noise" is minimized, and the results of the survey are more likely to replicate what is true in the population.
  • Larger samples increase the likelihood that the computed margin of error (卤3%, in this case) accurately reflects the possible range of true values of the parameter being measured, such as the percentage of households affected by crime.
Consequently, a thoughtfully chosen sample size helps in drawing conclusions that are both statistically significant and meaningful.
Binomial Proportion
A binomial proportion is a measure in statistics that represents the probability of a specified outcome in a binary situation, such as success or failure, yes or no, or in this case, whether or not a household experienced crime. In the crime survey:
  • The binomial proportion is the proportion of households that reported experiencing crime, which is 25% or 0.25.
  • It forms the basis for calculating other important statistics like the margin of error and confidence intervals.
  • For large enough sample sizes, like the one used in this survey (1002 households), the binomial proportion can also be quite accurately estimated.
In statistical terms, these calculations help us determine just how much of an estimate's variation is due to chance, guiding decision-makers in assessing trends and making informed statements about population behaviors from survey data.

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Most popular questions from this chapter

The article "National Geographic, the Doomsday Machine," which appeared in the March 1976 issue of the Journal of Irreproducible Results (yes, there really is a journal by that name -it's a spoof of technical journals!) predicted dire consequences resulting from a nationwide buildup of National Geographic magazines. The author's predictions are based on the observation that the number of subscriptions for National Geographic is on the rise and that no one ever throws away a copy of National Geographic. A key to the analysis presented in the article is the weight of an issue of the magazine. Suppose that you were assigned the task of estimating the average weight of an issue of National Geographic. How many issues should you sample to estimate the average weight to within \(0.1 \mathrm{oz}\) with \(95 \%\) confidence? Assume that \(\sigma\) is known to be 1 oz.

An article in the Chicago Tribune (August 29,1999 ) reported that in a poll of residents of the Chicago suburbs, \(43 \%\) felt that their financial situation had improved during the past year. The following statement is from the article: "The findings of this Tribune poll are based on interviews with 930 randomly selected suburban residents. The sample included suburban Cook County plus DuPage, Kane, Lake, McHenry, and Will Counties. In a sample of this size, one can say with \(95 \%\) certainty that results will differ by no more than 3 percent from results obtained if all residents had been included in the poll." Comment on this statement. Give a statistical argument to justify the claim that the estimate of \(43 \%\) is within \(3 \%\) of the true proportion of residents who feel that their financial situation has improved.

The article "Viewers Speak Out Against Reality TV" (Associated Press, September 12,2005 ) included the following statement: "Few people believe there's much reality in reality TV: a total of 82 percent said the shows are either 'totally made up' or 'mostly distorted'." This statement was based on a survey of 1002 randomly selected adults. Compute and interpret a bound on the error of estimation for the reported percentage.

The article "Students Increasingly Turn to Credit Cards" (San Luis Obispo Tribune, July 21,2006 ) reported that \(37 \%\) of college freshmen and \(48 \%\) of college seniors carry a credit card balance from month to month. Suppose that the reported percentages were based on random samples of 1000 college freshmen and 1000 college seniors. a. Construct a \(90 \%\) confidence interval for the proportion of college freshmen who carry a credit card balance from month to month. b. Construct a \(90 \%\) confidence interval for the proportion of college seniors who carry a credit card balance from month to month. c. Explain why the two \(90 \%\) confidence intervals from Parts (a) and (b) are not the same width.

Retailers report that the use of cents-off coupons is increasing. The Scripps Howard News Service (July 9 , 1991) reported the proportion of all households that use coupons as .77. Suppose that this estimate was based on a random sample of 800 households (i.e., \(n=800\) and \(p=.77\) ). Construct a \(95 \%\) confidence interval for \(\pi\), the true proportion of all households that use coupons.
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