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Chapter 8: Problem 19

Believe in ghosts A Harris poll of a random sample of 2303 adults in the United States in 2009 reported that \(82 \%\) believe in God, \(75 \%\) believe in heaven, \(61 \%\) believe in hell, and \(42 \%\) believe in ghosts. \(^{4}\) (Interestingly, these numbers are all substantially smaller than the corresponding values in a similar poll taken by Harris in \(2003 .\) ) The screen shot shows how the TI \(83+/ 84\) reports results of interval estimation at the \(95 \%\) confidence level for the proportion who believe in ghosts. Explain how to interpret the confidence interval shown.

Short Answer

Expert verified
The confidence interval suggests there is a 95% probability that the true proportion of adults in the US that believe in ghosts falls within this interval range.

Step by step solution

01

Understanding the Concept of Confidence Interval

A confidence interval is a range around a sample estimate that conveys where the true population parameter is likely to lie. If calculated correctly, and we were to take many samples, we expect approximately 95% of such confidence intervals to encompass the true population parameter.
02

Reviewing Confidence Interval Output

The TI 83+/84 calculator provides a confidence interval for the proportion; let's denote it as a range  \( [p_1, p_2] \) . This range estimates where the true proportion of the entire population who believe in ghosts is.
03

Interpretation of Confidence Interval

The range \( [p_1, p_2] \)  given by the calculator means that there is a 95% probability that the interval from  \( p_1 \)  to  \( p_2 \)  contains the true proportion  \( p \)  of all adults in the US who believe in ghosts. This implies a high level of confidence that the calculated proportion from our sample is a reliable estimator of the population proportion.

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Key Concepts

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

Proportion
Proportion is an essential concept in statistics and represents the part of a whole with relation to the total. In the context of our ghost belief example, the proportion we refer to is the percentage of people who believe in ghosts within the sampled population. For instance, if 42% of our sample believes in ghosts, then 0.42 represents the proportion of the sample that believes in this phenomenon. The importance of understanding proportion is crucial when making inferences about larger populations from random samples. Remember:
  • Proportion is usually expressed as a fraction, a percentage, or a decimal.
  • It provides a snapshot of the sample that can be extrapolated to infer characteristics of the broader population.
  • While the proportion itself is an estimate, understanding its context within a confidence interval offers a more informed prediction about the true population parameter.
Random Sample
A random sample is a critical component in obtaining reliable and unbiased estimators of the population parameters, like proportions. In our scenario, a random sample of 2303 U.S. adults was surveyed to determine beliefs in various paranormal phenomena, including ghosts. A random sample ensures:
  • Every individual in the population has an equal chance of being selected.
  • The results are generalizable to the entire population, reducing the likelihood of bias.
  • The sample better reflects the diversity and characteristics of the whole group.
Random sampling helps statisticians make assumptions about the broader population without needing to survey everyone. It makes the results more applicable to real-world settings while maintaining scientifically accepted accuracy. So next time you encounter a statistic, remember the power of a properly executed random sample in lending credibility and relevance to the data.
Population Parameter
Population parameter refers to a numerical characteristic or measure that defines an entire group of interest, known as the population. In the context of the poll about belief in ghosts, the population parameter of interest might be the actual proportion of all adults in the U.S. who believe in ghosts. While the true population parameter is often unknown, estimates derived from random samples help illustrate this concept:
  • Population parameters are fixed, though unknown in practice.
  • They include measures such as means, proportions, and variances.
  • Estimates from sample data serve as proxies to infer about these parameters.
By understanding population parameters, researchers can provide insights that go beyond the sample group, giving a picture of broader trends and characteristics that might apply universally.
Confidence Level
Confidence level is a statistical concept used to express how confident one is that a certain interval contains the true population parameter. In our example, a 95% confidence level was used when determining the existence of ghosts. Here are fundamental aspects of confidence levels:
  • A confidence level of 95% implies that if we were to take 100 random samples, approximately 95 of those intervals would contain the true population proportion.
  • A high confidence level means more certainty about the population parameters, though this comes at the cost of a wider interval.
  • It's crucial to balance the confidence level and interval width to draw practical conclusions.
Confidence levels help frame statistical estimates within a range that gives an assessment of accuracy, offering a vital tool in making informed decisions based on sample data.

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Most popular questions from this chapter

Length of hospital stays A hospital administrator wants to estimate the mean length of stay for all inpatients using that hospital. Using a random sample of 100 records of patients for the previous year, she reports that "The sample mean was \(5.3 .\) In repeated random samples of this size, the sample mean could be expected to fall within 1.0 of the true mean about \(95 \%\) of the time." Explain the meaning of this sentence from the report, showing what it suggests about the \(95 \%\) confidence interval.

U.S. popularity In \(2007,\) a poll conducted for the \(\mathrm{BBC}\) of 28,389 adults in 27 countries found that the United States had fallen sharply in world esteem since 2001 (www globescan.com). The United States was rated third most negatively (after Israel and Iran), with \(30 \%\) of those polled saying they had a positive image of the United States. a. In Canada, for a random sample of 1008 adults, \(56 \%\) said the United States is mainly a negative influence in the world. True or false: The \(99 \%\) confidence interval of (0.52,0.60) means that we can be \(99 \%\) confident that between \(52 \%\) and \(60 \%\) of the population of all Canadian adults have a negative image of the United States. b. In Australia, for a random sample of 1004 people, \(66 \%\) said the United States is mainly a negative influence in the world. True or false: The \(95 \%\) confidence interval of (0.63,0.69) means that for a random sample of 100 people, we can be \(95 \%\) confident that between 63 and 69 people in the sample have a negative image of the United States.

When the GSS recently asked subjects whether it should or should not be the government's responsibility to impose strict laws to make industry do less damage to the environment (variable GRNLAWS), 1403 of 1497 subjects said yes. a. What assumptions are made to construct a \(95 \%\) confidence interval for the population proportion who would say yes? Do they seem satisfied here? b. Construct the \(95 \%\) confidence interval. Interpret in context. Can you conclude whether or not a majority or minority of the population would answer yes?

How many businesses fail? A study is planned to estimate the proportion of businesses started in the year 2006 that had failed within five years of their start-up. How large a sample size is needed to guarantee estimating this proportion correct to within a. 0.10 with probability \(0.95 ?\) b. 0.05 with probability \(0.95 ?\) c. 0.05 with probability \(0.99 ?\) d. Compare sample sizes for parts a and \(\mathrm{b},\) and \(\mathrm{b}\) and \(\mathrm{c}\), and summarize the effects of decreasing the margin of error and increasing the confidence level.

Abstainers The Harvard study mentioned in the previous exercise estimated that \(19 \%\) of college students abstain from drinking alcohol. To estimate this proportion in your school, how large a random sample would you need to estimate it to within 0.05 with probability \(0.95,\) if before conducting the study a. You are unwilling to predict the proportion value at your school. b. You use the Harvard study as a guideline. c. Use the results from parts a and \(\mathrm{b}\) to explain why strategy (a) is inefficient if you are quite sure you'll get a sample proportion that is far from \(0.50 .\)
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