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Many sample surveys use well-designed random samples, but half or more of the original sample can't be contacted or refuse to take part. Any errors due to this nonresponse a. have no effect on the accuracy of confidence intervals. b. are included in the announced margin of error. c. are in addition to the random variation accounted for by the announced margin of error.

Short Answer

Expert verified
c. are in addition to the random variation accounted for by the announced margin of error.

Step by step solution

01

Understanding Nonresponse in Surveys

In sample surveys, nonresponse occurs when some individuals from the sample chosen do not participate. This nonresponse can introduce bias, meaning the survey results may not accurately represent the entire population.
02

Analyzing the Impact of Nonresponse

Nonresponse affects the accuracy of the survey results. When a significant proportion of the sample does not respond, the characteristics of nonrespondents may differ from respondents' characteristics, leading to biased results.
03

Examining the Margin of Error

The margin of error in a survey reflects the sampling error, which is the variability expected from taking a sample as opposed to surveying the whole population. It is computed assuming all sampled individuals have responded.
04

Relation of Nonresponse to Margin of Error

The announced margin of error does not cover errors due to nonresponse. It only accounts for random sampling variation, not systematic bias introduced by nonresponses. Errors due to nonresponse are in addition to the errors measured by the margin of error.
05

Conclusion

Given that nonresponse leads to potential biases not covered by the margin of error, errors due to nonresponse are in addition to the random variation that informs the announced margin of error.

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Key Concepts

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

Sample Surveys
Sample surveys are a popular method used in research to gather data about a large population. Instead of surveying every individual, which is often impractical, a subset, or sample, of the population is selected. This sample is expected to represent the larger group. To ensure that the sample reflects the diversity and characteristics of the overall population, researchers often use random sampling methods. Random sampling means every member of the population has an equal chance of being selected. Using a well-designed sample survey helps researchers collect data efficiently, saving time and resources while still obtaining reliable results. But, it is crucial to note that the accuracy heavily depends on the sample being unbiased and representative.
Margin of Error
The margin of error is a critical concept in statistics, especially when dealing with sample surveys. It is an indication of the possible error from using a sample instead of a full population. When results from a survey are presented, the margin of error gives a range that the true population parameter is likely to fall within. For example, if a survey result says 60% of people prefer coffee over tea, with a margin of error of ±5%, the truth might be anywhere from 55% to 65%. It accounts for random sampling errors but does not include all sources of error, such as measurement errors or biases from nonresponse.
Random Sampling Error
Random sampling error is the natural variation that occurs when a sample is taken. Because only a portion of the population is surveyed, there are differences between the sample results and the actual population characteristics. This type of error is predictable and measurable. It's the very reason why the margin of error exists—to provide an allowance for this variability in survey findings. However, random sampling error is different from errors caused by survey design or execution, such as nonresponse bias, which can skew results beyond what the margin of error covers.
Survey Bias
Survey bias refers to any systematic error in the survey process that results in an unrepresentative sample of the population. There are different kinds of biases that can affect surveys:
  • Selection bias: Occurs when certain groups are underrepresented because of the way the sample is chosen.
  • Nonresponse bias: Happens when the nonrespondents differ significantly in their opinions from those who do participate.
  • Response bias: Arises when the way questions are asked influences the responses given.
Each of these biases can lead to inaccurate survey results that do not truly reflect the views or characteristics of the entire population. It's crucial to minimize these biases during the design and implementation of a survey.

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Most popular questions from this chapter

Running Red Lights. A survey of licensed drivers inquired about running red lights. One question asked, "Of every 10 motorists who run a red light, about how many do you think will be caught?" The mean result for 880 respondents was \(x=1.92\), and the standard deviation was \(s=1.83 . \stackrel{2}{*}\) For this large sample, \(s\) will be close to the population standard deviation \(\sigma\), so suppose we know that \(\sigma=1.83 .\) a. Give a \(95 \%\) confidence interval for the mean opinion in the population of all licensed drivers. b. The distribution of responses is skewed to the right rather than Normal. This will not strongly affect the \(z\) confidence interval for this sample. Why not? c. The 880 respondents are an SRS from completed calls among 45,956 calls to randomly chosen residential telephone numbers listed in telephone directories. Only 5029 of the calls were completed. This information gives two reasons to suspect that the sample may not represent all licensed drivers. What are these reasons?

Multiple Testing. This problem assumes that you have studied optional Chaputer 14 on binomial distributions. If the null hypothesis is true, testing at significance level \(0.05\) means that the probability is \(0.05\) of incorrectly rejecting the null hypothesis. Suppose one conducts 20 independent tests at level \(0.05\) and in each case the null hypothesis is true. Let \(X\) denote the number of tests that incorrectly reject the null hypothesis. \(X\) can take values from 0 to 20 and will follow a binomial distribution with \(n=20\) observations and probability \(p=0.05\) of success. What is the probability \(X \geq 1\) ? This is the probability that at least one test will incorrectly reject the null hypothesis.

Sample Size and Margin of Error. Example 16.1 (page 368) described NHANES data on the body mass index (BMI) of 936 young men. The mean BMI in the sample was \(x=27.2 \mathrm{~kg} / \mathrm{m}^{2}\). We treated these data as an SRS from a Normally distributed population with standard deviation \(\sigma=11.6\) a. Suppose that we had an SRS of just 100 young men. What would be the margin of error for \(95 \%\) confidence? b. Find the margins of error for \(95 \%\) confidence based on SRSs of 400 young men and 1600 young men. c. Compare the three margins of error. How does increasing the sample size change the margin of error of a confidence interval when the confidence level and population standard deviation remain the same?

This Wine Stinks. How sensitive are the untrained noses of students? Exercise 16.27 (page 381) gives the lowest levels of dimethyl sulfide (DMS) that 10 students could detect. You want to estimate the mean DMS odor threshold among all students, and you would be satisfied to estimate the mean to within \(\pm 0.1\) with \(99 \%\) confidence. The standard deviation of the odor threshold for untrained noses is known to be \(\sigma=7\) micrograms per liter of wine. How large an SRS of untrained students do you need?

Type I and II Errors. Section 13.7 (page 311) discusses the prostatespecific antigen (PSA) test for prostate cancer. The test is not always correct, sometimes indicat ing prostate cancer (test is positive) when it is not present (a false positive) and often missing prostate cancer (test is negative) that is present (a false negative). Here is a table of the four possibilities. \begin{tabular}{|l|l|l|} \hline & Test Result \\ \hline Cancer present & Positive & Negative \\ \hline Cancer absent & False positive & False negative \\ \hline \end{tabular} If we treat "Cancer absent" as our null hypothesis and the PSA test result as our test statistic, which of the four combinations corresponds to a Type I error? Which corresponds to a Type II error?

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