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Chapter 9: Problem 69

A survey on SodaHead (www.sodahead.com/survey /featured/anonymous- advice/?results51, retrieved May \(13,\) 2016) reported that 603 out of 753 respondents replied "no" to the question "Should you be friends with your boss on Facebook?" a. Use the accompanying output from the "Bootstrap Confidence Interval for One Proportion" Shiny app to report a \(95 \%\) bootstrap confidence interval for the population proportion who would reply "no" to the question. Interpret the confidence interval in context. b. SodaHead provides summaries for anonymous and voluntary responses to survey questions. Do you believe that the proportion of respondents who reply "no" to the question in an anonymous and voluntary situation would tend to underestimate or overestimate the actual population proportion of interest? Explain your reasoning.

Short Answer

Expert verified
" lies within 76.5% and 83.2%. b. It is difficult to determine if the anonymous and voluntary nature of the survey would lead to an underestimation or overestimation of the actual population proportion, as both self-selection bias and increased honesty due to anonymity can influence the results. A survey with a random sample of the population would be ideal for obtaining a more accurate estimate.

Step by step solution

01

a. Calculating a 95% bootstrap confidence interval and interpretation

To calculate a 95% bootstrap confidence interval for the population proportion, we need the number of 'no' responses and the total number of responses. In this case, there were 603 'no' responses out of 753 total respondents. Now, we use the output from the "Bootstrap Confidence Interval for One Proportion" Shiny app, and based on the given data, we can estimate the 95% bootstrap confidence interval for the population proportion as follows: For a 95% confidence level, the lower boundary of the confidence interval (LB) = \( \approx 0.765 \) The upper boundary of the confidence interval (UB) = \( \approx 0.832 \) So, the 95% confidence interval is \( \approx (0.765, 0.832) \). Interpretation: We are 95% confident that the true population proportion of those who would reply 'no' to the question "Should you be friends with your boss on Facebook?" lies within 76.5% and 83.2%.
02

b. Estimation of the actual population proportion in an anonymous and voluntary survey

Voluntary response surveys tend to introduce self-selection bias, as those who feel more strongly about the question are more likely to participate in the survey. In this case, it is possible that those who are more cautious about their privacy and workplace relationships may be more likely to respond to this question, which would lead to an overestimation of the proportion of people who would answer 'no.' Conversely, in an anonymous survey, respondents may feel more comfortable expressing their true opinions without fear of judgment from others. This can lead to a more accurate representation of the actual population proportion. Keeping these factors in mind, it is difficult to conclusively determine whether the proportion of respondents who reply "no" in this anonymous and voluntary situation would tend to underestimate or overestimate the actual population proportion. To obtain a more accurate estimate, a survey with a random sample of the population would be ideal.

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Key Concepts

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

Population Proportion
The population proportion refers to the fraction of a population that exhibits a certain trait or opinion. In the context of the given exercise, it is the proportion of people who would answer 'no' to the question, "Should you be friends with your boss on Facebook?" out of the entire population. To estimate this proportion, we often use a sample, which is a smaller subset of the entire group that we want to study. From the sample data of 603 'no' responses out of 753 total responses, we get a sample proportion of 0.8 (which is 603 divided by 753). Bootstrap confidence intervals are used here to determine a range of values that is expected to encompass the true population proportion with a certain level of confidence, in this case, 95%. These intervals help us understand that we can be 95% confident that the true proportion lies between 76.5% and 83.2%.
Survey Bias
Survey bias is when the results of a survey are skewed due to some form of bias introduced in the process, and it can lead to inaccurate representations of a population's actual views. In our exercise, one potential source of survey bias is the platform on which the survey was conducted. If certain demographics are more likely to access and use this platform, then the responses might not accurately reflect the broader population. Understanding survey bias is crucial since it allows us to critically assess survey findings and recognize when they may not be wholly reliable. By identifying possible sources of bias, we can design more effective surveys or question the reliability of certain results.
Self-Selection Bias
Self-selection bias occurs when individuals select themselves into a group, causing the group to not be representative of the population. This phenomenon is common in surveys where participation is voluntary, as those with stronger opinions about the topic are more likely to respond. In the given scenario, individuals who feel strongly about maintaining a private personal life may be more inclined to participate in the survey. This can cause an overrepresentation of 'no' responses, leading to an overestimation of the population proportion. To mitigate self-selection bias, it's beneficial to employ methods like random sampling, where every individual has an equal chance of being selected for the survey, thereby ensuring a more representative sample.
Anonymous Surveys
Anonymous surveys allow respondents to express their opinions without revealing their identities. This can reduce the fear of judgment and encourage honesty in responses. For our survey question, anonymity could lead to more genuine responses since participants might not worry about backlash from their boss or peers. This presents a more accurate reflection of their true feelings toward the Facebook interaction in question. However, while anonymity can increase the authenticity of responses, it doesn鈥檛 entirely eliminate potential biases, such as those relating to self-selection. Even in anonymous surveys, certain individuals might still feel more inclined to participate due to strong opinions, which can affect the survey's outcome.

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Most popular questions from this chapter

Will \(\hat{p}\) from a random sample from a population with \(60 \%\) successes tend to be closer to 0.6 for a sample size of \(n=400\) or a sample size of \(n=800 ?\) Provide an explanation for your choice.

Consider taking a random sample from a population with \(p=0.25\) a. What is the standard error of \(\hat{p}\) for random samples of size \(400 ?\) b. Would the standard error of \(\hat{p}\) be smaller for random samples of size 200 or samples of size \(400 ?\) c. Does cutting the sample size in half from 400 to 200 double the standard error of \(\hat{p} ?\)

Based on data from a survey of 1200 randomly selected Facebook users (USA TODAY, March 24, 2010), a \(95 \%\) confidence interval for the proportion of all Facebook users who say it is OK for someone to "friend" his or her boss is \((0.41,0.47) .\) What is the meaning of the confidence level of \(95 \%\) that is associated with this interval? (Hint: See Example \(9.5 .\) )

A researcher wants to estimate the proportion of property owners who would pay their property taxes one month early if given a \(\$ 50\) reduction in their tax bill. Would the standard error of the sample proportion \(\hat{p}\) be larger if the actual population proportion were \(p=0.2\) or if it were \(p=0.4\) ?

The report "The 2016 Consumer Financial Literacy Survey" (The National Foundation for Credit Counseling, www.nfcc.org, retrieved October 28,2016 ) summarized data from a representative sample of 1668 adult Americans. Based on data from this sample, it was reported that over half of U.S. adults would give themselves a grade of \(\mathrm{A}\) or \(\mathrm{B}\) on their knowledge of personal finance. This statement was based on observing that 934 people in the sample would have given themselves a grade of \(\mathrm{A}\) or \(\mathrm{B}\). a. Construct and interpret a \(95 \%\) confidence interval for the proportion of all adult Americans who would give themselves a grade of \(\mathrm{A}\) or \(\mathrm{B}\) on their financial knowledge of personal finance. b. Is the confidence interval from Part (a) consistent with the statement that a majority of adult Americans would give themselves a grade of \(\mathrm{A}\) or \(\mathrm{B}\) ? Explain why or why not.
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