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Chapter 8: Problem 63

British monarchy In February 2002, the Associated Press quoted a survey of 3000 British residents conducted by YouGov.com. It stated, "Only \(21 \%\) wanted to see the monarchy abolished, but \(53 \%\) felt it should become more democratic and approachable. No margin of error was given." If the sample was random, find the \(95 \%\) margin of error for each of these estimated proportions.

Short Answer

Expert verified
For the abolition: 1.46%; for democratization: 1.78%.

Step by step solution

01

Understanding Margin of Error Formula

The margin of error (MOE) for a proportion is calculated using the formula: \( MOE = z \times \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} \), where \( z \) is the z-score corresponding to the desired level of confidence, \( \hat{p} \) is the sample proportion, and \( n \) is the sample size. For a 95% confidence level, the z-score is typically 1.96.
02

Calculate Margin of Error for Abolition of Monarchy

Given that \( \hat{p} = 0.21 \) (21%) and \( n = 3000 \), we'll use the formula to find the margin of error for the proportion wanting to abolish the monarchy: \[ MOE = 1.96 \times \sqrt{\frac{0.21 \times (1-0.21)}{3000}} \] Calculating inside the square root gives:\[ \frac{0.21 \times 0.79}{3000} = \frac{0.1659}{3000} \approx 0.0000553 \]So the margin of error is:\[ MOE \approx 1.96 \times \sqrt{0.0000553} \approx 1.96 \times 0.00743 \approx 0.0146 \]The margin of error for this proportion is approximately \(0.0146\) or 1.46%.
03

Calculate Margin of Error for Democratisation of Monarchy

For the proportion feeling the monarchy should become more democratic \( (\hat{p} = 0.53) \): \[ MOE = 1.96 \times \sqrt{\frac{0.53 \times (1-0.53)}{3000}} \] Calculating inside the square root gives:\[ \frac{0.53 \times 0.47}{3000} = \frac{0.2491}{3000} \approx 0.00008303 \]So the margin of error is:\[ MOE \approx 1.96 \times \sqrt{0.00008303} \approx 1.96 \times 0.00911 \approx 0.0178 \]The margin of error for this proportion is approximately \(0.0178\) or 1.78%.

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Key Concepts

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

Margin of Error
In statistics, the margin of error is a measurement of the potential variation between the sampled population and the actual population. Imagine you're trying to estimate how a whole group of people feels about something, based on asking just a few of them. This is where the margin of error becomes handy, helping to gauge how close our survey results might be to the actual figures if we surveyed everyone.
  • The formula for calculating the margin of error when dealing with proportions is crucial. It is: \( MOE = z \times \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} \). This formula involves three key factors 鈥� the z-score, the sample proportion \( \hat{p} \), and the sample size \( n \).
  • The z-score corresponds to your desired confidence level 鈥� in a 95% confidence level, it's typically 1.96. Think of the z-score as telling us just how sure we are about the results essentially.
  • The sample proportion \( \hat{p} \) is the fraction of the sample displaying the characteristic of interest, such as 21% wanting to abolish the monarchy in our example. Meanwhile, \( n \) is the total number of observations or people surveyed 鈥� 3000 in the dataset used.
This concept is extremely valuable as it offers insights into the reliability and accuracy of the survey's estimations.
Survey Analysis
Survey analysis serves as the backbone for understanding opinions, behaviors, and characteristics of a population. By conducting a survey, researchers can draw important conclusions about a group by asking a representative sample a set of standardized questions.
  • The key advantage of survey analysis lies in its ability to gather a large quantity of data quickly and relatively inexpensively. This makes it appealing for assessing public opinion on topics like governmental policies or social issues.
  • However, surveys must be designed carefully to ensure that the questions are clear, unbiased, and structured. This ensures the data collected is valid and reflects the actual sentiments of the populace.
  • In our British monarchy example, survey analysis highlighted conflicting viewpoints: a minority desiring abolition versus a majority seeking democratization. Without a proper margin of error, results might be misleading, emphasizing the importance of complete data presentation.
Thus, survey analysis, when accurately structured, guides insightful decision-making and helps researchers and authorities understand public trends.
Confidence Level
The confidence level in statistics indicates how certain we are that the surveyed sample accurately represents the general population. It's expressed as a percentage, with higher percentages indicating greater confidence.
  • When we discuss a 95% confidence level, this means we can be 95% certain that the survey results fall within the margin of error. Practically, this is akin to saying, "If we conducted this survey 100 times, the results would be the same in 95 cases."
  • The chosen confidence level impacts the z-score used in the margin of error calculation. A higher confidence level requires a larger z-score, hence a wider margin of error, reflecting increased certainty but acknowledging greater potential variance.
  • This balance is crucial: Selecting an extremely high confidence level can yield impractically broad estimates, whereas too low a level may produce misleading precision. For the British monarchy survey, utilizing a 95% level offers a balanced approach of reliability and precision.
Ultimately, confidence levels are fundamental for assessing the global applicability of survey findings.
Proportion Estimation
Proportion estimation is a vital statistical tool used to deduce how a certain characteristic is distributed within a population. When estimating proportions, surveys identify the fraction or percentage of a population that exhibits a specific trait.
  • For example, in regards to the British monarchy survey, the estimation involves calculating the proportion believing in abolition (21%) and increased democratization (53%).
  • When predicting proportions, the formula \( \hat{p} = \frac{x}{n} \) becomes relevant, where \(x\) is the number of people with the trait, and \(n\) is the total surveyed population.
  • Importantly, accurate proportion estimation hinges on survey samples reflecting the diversity of the entire population. When the sample is biased or unrepresentative, the conclusions drawn about proportions may be significantly skewed.
In summary, proportion estimation sheds light on the distribution of opinions or characteristics and plays a vital role in helping researchers and decision-makers understand broader societal trends.

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Most popular questions from this chapter

For the number of hours of TV watching, the 2008 GSS reported a mean of 2.98 for the 1324 white subjects, with a standard deviation of \(2.66 .\) The mean was 4.38 for the 188 black subjects, with a standard deviation of 3.58 . Analyze these data, preparing a short report in which you mention the methods used and the assumptions on which they are based, and summarize and interpret your findings.

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?

Types of estimates An interval estimate for a mean is more informative than a point estimate, because with an interval estimate you can figure out the point estimate, but with the point estimate alone you have no idea how wide the interval estimate is. Explain why this statement is correct, illustrating using the reported \(95 \%\) confidence interval of (4.0,5.6) for the mean number of dates in the previous month based on a sample of women at a particular college.

How often feel sad? Refer to the previous exercise. This question was asked of 10 students in a class at the University of Wisconsin recently. The responses were $$ 0,0,1,0,7,2,1,0,0,3 $$ Find and interpret a \(90 \%\) confidence interval for the population mean, and indicate what you would have to assume for this inference to apply to the population of all University of Wisconsin students.

Median as point estimate When the population distribution is normal, the population mean equals the population median. How good is the sample median as a point estimate of this common value? For a random sample, the estimated standard error of the sample median equals \(1.25(s / \sqrt{n}) .\) If the population is normal, explain why the sample mean tends to be a better estimate than the sample median.
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