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Chapter 9: Problem 30

Explaining confidence (8.2) Here is an explanation from a newspaper concerning one of its opinion polls. Explain what is wrong with the following statement. For \(a\) poll of 1,600 adults, the variation due to sampling error is no more than three percentage points either way. The error margin is said to be valid at the 95 percent confidence level. This means that, if the same questions were repeated in 20 polls, the results of at least 19 surveys would be within three percentage points of the results of this survey.

Short Answer

Expert verified
The statement misinterprets the confidence level related to repeating the same poll 20 times; it actually applies to independent, different polls.

Step by step solution

01

Understanding Polling Error

The statement mentions a sampling error of no more than three percentage points with a 95% confidence level. This implies that the margin of error is expected when interpreting the results of a poll.
02

Interpreting Confidence Levels

A 95% confidence level indicates that if the survey were repeated many times, we expect the true population parameter (e.g., true proportion of an opinion) to fall within the margin of error in 95% of those polls.
03

Identifying the Fault

The error in the statement lies in how the repetition of 20 polls is described. It suggests that the specific result of one poll will fall within the margin in at least 19 out of 20 repeated polls, which is a misuse of the confidence interval concept.
04

Clarifying the Misinterpretation

The correct interpretation is that if the polling process itself were repeated independently 20 times, approximately 19 times the estimate of the polling would be accurate within the margin. It does not apply to a single poll being repeated.

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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 statistical term that represents the range within which we expect the true value of a population parameter to lie, based on the results of a survey or poll.
This margin helps account for small variations caused by the sampling process.
For instance, if a poll indicates that 60% of people support a policy with a margin of error of 3%, this means the true percentage of support is likely between 57% and 63%. Understanding the margin of error is crucial because it gives context to the poll results.
  • Determines Precision: A smaller margin of error indicates more precise survey results.
  • Dependent on Sample Size: Larger samples usually result in a smaller margin of error.
  • Not the Same as Sampling Error: While related, margin of error specifically involves the range of possible results due to sampling.
In practice, the margin of error is paired with a confidence level to describe the reliability of poll results.
Sampling Error
Sampling error happens because we are looking at a sample, or a smaller part of the whole population, instead of the full population itself. No sample is a perfect representation.
Therefore, every poll or survey includes a sampling error. This error arises simply from having observations from only a part of the total population.
This means the results might slightly differ from what you would find if you questioned everyone.
  • Not a Problem, But an Inevitability: Sampling error isn't "wrong"; it's a natural part of using samples.
  • Managed by Proper Sampling: Using random sampling methods can help reduce sampling error.
  • Adjustable with Sample Size: Bigger samples generally decrease the sampling error.
Understanding the role of sampling error is foundational in assessing the accuracy of survey or poll results.
Confidence Level
The confidence level indicates the likelihood that the results of a poll reflect the true opinions or characteristics of the entire population. It is expressed as a percentage, commonly 95% or 99%.
A high confidence level means you can be "confident" in the results.
This doesn't guarantee accuracy for individual trials, but it offers an overall assurance of reliability. A 95% confidence level, for example, suggests that if you conducted many polls, you would expect the true value to fall within the margin of error in 95 out of 100 of those trials.
  • Association with Margin of Error: Together, these tools create a picture of where the true value might lie.
  • Improving Confidence: Often influenced by sample size, with larger samples providing higher confidence levels.
  • Key for Decision-Making: Confidence levels help make informed decisions based on data from surveys.
Understanding the confidence level is essential for interpreting polls and trusting their findings.

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Most popular questions from this chapter

Vigorous exercise helps people live several years longer (on average). Whether mild activities like slow walking extend life is not clear. Suppose that the added life expectancy from regular slow walking is just 2 months. A statistical test is more likely to find a significant increase in mean life expectancy if (a) it is based on a very large random sample and a \(5 \%\) significance level is used. (b) it is based on a very large random sample and a \(1 \%\) significance level is used. (c) it is based on a very small random sample and a \(5 \%\) significance level is used. (d) it is based on a very small random sample and a \(1 \%\) significance level is used. (e) the size of the sample doesn't have any effect on the significance of the test.

A drug manufacturer claims that fewer than \(10 \%\) of patients who take its new drug for treating Alzheimer's disease will experience nausea. To test this claim, a significance test is carried out of $$ \begin{array}{l} H_{0}: p=0.10 \\ H_{a}: p<0.10 \end{array} $$ You learn that the power of this test at the \(5 \%\) significance level against the alternative \(p=0.08\) is 0.29 . (a) Explain in simple language what "power \(=0.29 "\) means in this setting. (b) You could get higher power against the same alternative with the same \(\alpha\) by changing the number of measurements you make. Should you make more measurements or fewer to increase power? Explain. (c) If you decide to use \(\alpha=0.01\) in place of \(\alpha=0.05\), with no other changes in the test, will the power increase or decrease? Justify your answer. (d) If you shift your interest to the alternative \(p=0.07\) with no other changes, will the power increase or decrease? Justify your answer.

In the "Ask Marilyn" column of Parade magazine, a reader posed this question: "Say that a slot machine has five wheels, and each wheel has five symbols: an apple, a grape, a peach, a pear, and a plum. I pull the lever five times. What are the chances that I'll get at least one apple?" Suppose that the wheels spin independently and that the five symbols are equally likely to appear on each wheel in a given spin. (a) Find the probability that the slot player gets at least one apple in one pull of the lever. Show your method clearly. (b) Now answer the reader's question. Show your method clearly.

Exercises 21 and 22 refer to the following setting. Slow response times by paramedics, firefighters, and policemen can have serious consequences for accident victims. In the case of life-threatening injuries, victims generally need medical attention within 8 minutes of the accident. Several cities have begun to monitor emergency response times. In one such city, the mean response time to all accidents involving life-threatening injuries last year was \(\mu=6.7\) minutes. Emergency personnel arrived within 8 minutes on \(78 \%\) of all calls involving life-threatening injuries last year. The city manager shares this information and encourages these first responders to "do better." At the end of the year, the city manager selects an SRS of 400 calls involving life-threatening injuries and examines the response times. (a) State hypotheses for a significance test to determine whether the average response time has decreased. Be sure to define the parameter of interest. (b) Describe a Type I error and a Type II error in this setting, and explain the consequences of each. (c) Which is more serious in this setting: a Type I error or a Type II error? Justify your answer.

. You are testing \(H_{0}: \mu=10\) against \(H_{a}: \mu<10\) based on an SRS of 20 observations from a Normal population. The \(t\) statistic is \(t=-2.25\). The \(P\) -value (a) falls between 0.01 and 0.02 . (b) falls between 0.02 and 0.04 . (c) falls between 0.04 and 0.05 . (d) falls between 0.05 and 0.25 . (e) is greater than 0.25 .
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