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Chapter 8: Problem 96

Changing views of United States The June 2003 report on Views of a Changing World, conducted by the Pew Global Attitudes Project (www.people-press.org), discussed changes in views of the United States by other countries. In the largest Muslim nation, Indonesia, a poll conducted in May 2003 after the Iraq war began reported that \(83 \%\) had an unfavorable view of America, compared to \(36 \%\) a year earlier. The 2003 result was claimed to have a margin of error of 3 percentage points. How can you approximate the sample size the study was based on?

Short Answer

Expert verified
The study's sample size was approximately 597.

Step by step solution

01

Understand the Problem

We need to approximate the sample size of a study about the views of Indonesians on the United States, given the unfavorable view was 83% in May 2003, with a margin of error of 3 percentage points.
02

Define the Margin of Error Formula

The margin of error (ME) for a proportion is calculated as follows: \( ME = z \times \sqrt{\frac{p(1-p)}{n}} \), where \( z \) is the z-score corresponding to the confidence level, \( p \) is the proportion (0.83 in this case), and \( n \) is the sample size.
03

Identify the Confidence Level and Z-score

For this problem, we'd typically assume a 95% confidence level, where the z-score \( z \approx 1.96 \). This is standard unless otherwise specified.
04

Rearrange the Formula to Solve for Sample Size

Rearrange the margin of error formula to solve for \( n \): \( n = \frac{(z^2 \times p \times (1-p))}{ME^2} \).
05

Substitute Known Values into the Formula

Substituting, we have \( n = \frac{(1.96^2 \times 0.83 \times 0.17)}{0.03^2} \).
06

Calculate the Sample Size

First, calculate \( 1.96^2 = 3.8416 \), then substitute to get \( n = \frac{(3.8416 \times 0.83 \times 0.17)}{0.0009} \). This evaluates to approximately \( n \approx 597.01 \).
07

Approximate to the Nearest Whole Number

Since sample size must be a whole number, round \( n \) to the nearest whole number, which is 597.

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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 a statistic that quantifies the amount by which our estimates might be off from the true population value. It provides us with a range that expresses an uncertainty level around the sample result. For example, if a survey of Indonesians showed 83% had an unfavorable view of America with a margin of error of 3 percentage points, it means the true proportion of Indonesians with an unfavorable view is likely between 80% and 86%.

Margin of error depends on three factors:
  • The sample size: Larger sample sizes yield smaller margins of error, as they represent the population more accurately.
  • The confidence level: Higher confidence levels increase the margin of error because they demand more certainty in capturing the true population parameter.
  • The variability of the data: Greater variability leads to larger margins of error, as the spread of data requires broader estimates.
To calculate the margin of error for proportions, we use this formula: \( ME = z \times \sqrt{\frac{p(1-p)}{n}} \), where \( p \) is the sample proportion and \( n \) is the sample size.
Confidence Level
The confidence level is a measure of how confident we can be in the results of our survey, expressing the percentage of all possible samples that can be expected to include the true population parameter.

Typically, a 95% confidence level is used, which implies that if the survey were to be repeated many times, 95% of the calculated confidence intervals would include the true population proportion. This standard confidence level is often associated with a z-score of approximately 1.96. The choice of confidence level affects the margin of error:
  • Higher confidence levels, such as 99%, require a larger margin of error or a larger sample size to maintain precision.
  • Lower confidence levels, such as 90%, allow for a smaller sample size but yield less certainty about the interval including the true population parameter.
Confidence levels are chosen based on the level of certainty desired in the results and the resources available for data collection.
Proportion Calculations
Proportion calculations are essential in statistics when dealing with categorical data, such as survey results or polls. Here, we use the proportion of the sample to draw inferences about the population. In the exercise regarding Indonesia's view of America, the proportion (\( p \)) was 0.83, reflecting the sample in which 83% viewed America unfavorably.

Calculating and understanding proportions is crucial when:
  • Estimating population parameters from sample data, which helps in making predictions or decisions based on survey findings.
  • Comparing differences between sub-groups, for instance, viewing proportions of different demographic groups for insights.
  • Evaluating changes over time, by comparing proportions from sequential surveys to observe trends or shifts in opinion.
For accurate proportion calculations, we use the formula \( n = \frac{(z^2 \times p \times (1-p))}{ME^2} \), where \( n \) is necessary to ensure the margin of error stays within the defined acceptable limits.

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Most popular questions from this chapter

Driving after drinking In December \(2004,\) a report based on the National Survey on Drug Use and Health estimated that \(20 \%\) of all Americans of ages 16 to 20 drove under the influence of drugs or alcohol in the previous year (AP, December 30,2004 ). A public health unit in Wellington, New Zealand, plans a similar survey for young people of that age in New Zealand. They want a \(95 \%\) confidence interval to have a margin of error of 0.04 . a. Find the necessary sample size if they expect results similar to those in the United States. b. Suppose that in determining the sample size, they use the safe approach that sets \(\hat{p}=0.50\) in the formula for \(n\). Then, how many records need to be sampled? Compare this to the answer in part a. Explain why it is better to make an educated guess about what to expect for \(\hat{p},\) when possible.

Effect of \(n\) Find the margin of error for a \(95 \%\) confidence interval for estimating the population mean when the sample standard deviation equals 100 , with a sample size of (i) 400 and (ii) 1600 . What is the effect of the sample size?

Income of Native Americans How large a sample size do we need to estimate the mean annual income of Native Americans in Onondaga County, New York, correct to within $$\$ 1000$$ with probability \(0.99 ?\) No information is available to us about the standard deviation of their annual income. We guess that nearly all of the incomes fall between $$\$ 0$$ and $$\$ 120,000$$ and that this distribution is approximately bell shaped.

Health care A study dealing with health care issues plans to take a sample survey of 1500 Americans to estimate the proportion who have health insurance and the mean dollar amount that Americans spent on health care this past year. a. Identify two population parameters that this study will estimate. b. Identify two statistics that can be used to estimate these parameters.

Believe in heaven? When a GSS asked 1326 subjects, "Do you believe in heaven?" (coded HEAVEN), the proportion who answered yes was \(0.85 .\) From results in the next section, the estimated standard deviation of this point estimate is 0.01 . a. Find and interpret the margin of error for a \(95 \%\) confidence interval for the population proportion of Americans who believe in heaven. b. Construct the \(95 \%\) confidence interval. Interpret it in context.
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