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Bias in sampling

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Difficulty: 1 – 5

54666112

At a crowded festival, an interviewer surveys the nearest person whenever a previous interview ends. Identify the sampling method and explain why it may be biased.

Hints

- Ask whether chance determines who is selected. - Identify which visitors are easiest for the interviewer to reach. - Explain why accessibility could be related to visitor characteristics.

Solution

1. The interviewer chooses whoever is easiest to reach. 2. This is convenience sampling, not random sampling. 3. People standing near the interview location or moving slowly are more likely to be included. 4. Those people may differ from the full festival population.

Answer

It is a convenience sample. Selection depends on proximity and timing, so some festival visitors are much more likely to be included than others.

55626612

Classify the main problem in each survey as nonresponse or voluntary response. a) A county randomly selects \(300\) residents and mails them a survey. Ninety selected residents never return it. b) A radio station posts a poll on its website and invites any visitor to answer.

Hints

- For each case, ask whether people were selected first and then failed to answer. - Contrast that with a survey in which people choose for themselves whether to join the sample.

Solution

1. In a), the residents were selected for the sample but some did not provide data. That is nonresponse. 2. In b), people decide for themselves whether to enter the sample. That is voluntary response.

Answer

a) Nonresponse. b) Voluntary response.

54654612

A town survey asks, “Do you support the responsible plan to protect families by adding more streetlights?” Explain the likely source of bias and rewrite the item to reduce it.

Hints

- Identify words that praise one response or make the alternative sound undesirable. - Preserve the policy choice while removing emotional language. - Keep the revised response options balanced.

Solution

1. Words such as “responsible” and “protect families” favor one response. 2. This creates question wording response bias. 3. A neutral version asks directly about support without persuasive language.

Answer

The wording is leading and can create response bias toward support. A less biased item is: “Do you support or oppose the proposal to add more streetlights in the town?”

54655612

Students complete a school survey that asks them to report how many energy drinks they consumed during the previous week. Their names are printed at the top, and coaches can view the responses. Identify the likely bias and explain a plausible direction.

Hints

- Consider whether respondents have a reason to conceal the true value. - Identify who can connect answers to individuals. - Use the likely reporting behavior to determine the direction.

Solution

1. The sensitive behavior is self-reported without anonymity. 2. Students may report socially desirable answers because coaches can identify them. 3. This is response bias and will likely make reported consumption lower than actual consumption.

Answer

Response bias is likely. Because the survey is identifiable and coaches can see it, some students may underreport energy-drink use, making the estimated mean consumption too low.

54657112

Police officers conduct face-to-face interviews outside a station and ask residents whether they have ever ignored a minor traffic law. Identify the likely source of bias and describe how the interview setting could affect responses.

Hints

- Consider whether the respondent may feel judged by the interviewer. - Compare a truthful answer with a socially safer answer. - Use that pressure to determine a likely direction.

Solution

1. The question concerns behavior that respondents may not want to admit. 2. The presence of police officers can influence answers away from the truth. 3. This is response bias caused by the interviewer and social desirability. 4. Residents will likely underreport ignoring traffic laws.

Answer

Response bias is likely. Interviewing residents face-to-face with police officers may lead them to underreport behavior that violates traffic laws.

54659112

A survey asks adults, “Do you exercise regularly?” Explain why the wording can create response problems and rewrite the item so answers are more comparable.

Hints

- Identify the word that different respondents may define differently. - Add a clear time period. - Ask for a measurable quantity rather than a vague label.

Solution

1. The word “regularly” has no fixed frequency or duration. 2. Respondents may interpret it differently, creating measurement and response bias. 3. A better item defines the activity, time period, and quantity to report.

Answer

The term “regularly” is ambiguous. One improved item is: “During the past \(7\) days, on how many days did you do at least \(30\,\text{minutes}\) of moderate or vigorous physical activity?”

54660112

A customer-service survey asks, “How would you rate our outstanding support?” The response options are “Excellent,” “Very good,” and “Good.” Identify two wording or response-option problems and explain their likely effect.

Hints

- Examine both the question stem and the available choices. - Ask whether a dissatisfied respondent has an accurate option. - Determine which direction the wording encourages.

Solution

1. The word “outstanding” is leading and favors a positive response. 2. All response options are positive, so dissatisfied customers cannot answer accurately. 3. These features create response bias and will tend to make support ratings appear more favorable than they are.

Answer

The question is leading, and the response scale omits neutral and negative choices. Both problems will likely bias ratings upward.

54664112

A survey asks shoppers to report the exact amount they spent on restaurant meals during the previous twelve months. Identify a likely response problem and suggest a less biased measurement approach.

Hints

- Consider how accurately a person can reconstruct the requested value. - Identify whether errors are likely to arise from memory. - Reduce the memory burden or use recorded data.

Solution

1. Recalling exact spending over twelve months is difficult. 2. Memory errors create response bias or measurement error. 3. A shorter recall period, spending records, or a diary would produce more accurate data.

Answer

The question is vulnerable to recall-related response bias. Use a shorter period such as the previous week or month, or ask respondents to consult records or keep a spending diary.

54665112

A campus survey asks, “Are you satisfied with the dining hall's food quality and operating hours?” Explain why the item is problematic and rewrite it.

Hints

- Count the distinct topics joined in the question. - Consider a respondent who likes one feature but dislikes the other. - Separate the topics while keeping the response scales comparable.

Solution

1. The item asks about two distinct features in one question. 2. A respondent may have different opinions about food quality and hours. 3. One answer cannot reveal which feature caused the response. 4. Split the item into two neutrally worded questions.

Answer

The item is double-barreled. Use separate questions, such as “How satisfied are you with the dining hall's food quality?” and “How satisfied are you with the dining hall's operating hours?”

54665612

Immediately after a televised political debate, a network invites viewers to text one of two numbers to vote for the winner. Explain why the result should not be reported as the preference of all debate viewers.

Hints

- Identify who controls entry into the poll. - Consider which viewers have the strongest motivation to text. - Compare respondents with the full audience.

Solution

1. Viewers choose whether to participate. 2. Those with strong opinions, greater enthusiasm, or easier access to texting may be more likely to respond. 3. This is voluntary response bias. 4. The respondents are not a random sample of all viewers.

Answer

The text poll is a voluntary response sample. Highly motivated viewers are overrepresented, so the result cannot represent all debate viewers.

54666612

A cafeteria measures food waste on days when students can see staff members standing beside the disposal bins with clipboards. Explain how the observation procedure could bias the measured waste.

Hints

- Ask whether people know they are being observed. - Consider how that awareness could change the target behavior. - Compare observed behavior with normal behavior.

Solution

1. Students know their disposal behavior is being observed. 2. They may alter behavior by taking less food, hiding waste, or disposing of it elsewhere. 3. The measured behavior may differ systematically from usual behavior. 4. This observation-related response bias will likely make recorded waste too low.

Answer

Visible observation may cause students to change their behavior, creating response bias. Recorded cafeteria waste will likely be lower than usual waste.

54671112

A software company emails a satisfaction survey to every customer but analyzes only the customers who choose to respond. A manager calls the results a census because everyone received the invitation. Evaluate that claim and identify the primary missing-data problem.

Hints

- Distinguish inviting every population member from obtaining data from every population member. - Classify people who were contacted but supplied no response. - Ask when the missing responses would systematically affect the satisfaction estimate.

Solution

1. A census requires data from every member of the population, not merely an invitation sent to every member. 2. Many contacted customers provide no data, so the analyzed responses do not constitute a census. 3. The primary problem is nonresponse: population members were contacted but did not respond. 4. The results can be biased if responding customers differ systematically in satisfaction from nonresponding customers.

Answer

The results are not a census because data were not obtained from every customer. The primary problem is nonresponse, and the satisfaction estimate may be biased if respondents differ systematically from customers who did not answer.

54674612

Immediately after resolving a customer's complaint, the same service agent asks the customer to rate the agent's performance aloud. Explain the likely response bias and suggest a less biased procedure.

Hints

- Consider how the respondent may feel when the person being evaluated is present. - Ask which direction politeness pressure would move the rating. - Separate the evaluator from the response-collection process.

Solution

1. The customer may feel pressure to avoid criticizing the agent directly or may answer out of politeness or gratitude. 2. This social interaction can systematically inflate ratings. 3. The rating should be collected privately through an anonymous or confidential system after the interaction, without the agent seeing or recording the response.

Answer

The face-to-face request can create courtesy or social-desirability bias, likely pushing ratings upward. Collect the rating privately and confidentially through a separate system.

54679912

An online survey asks whether users want to renew a service. The option “Yes, renew automatically” is preselected, and users must click another option to change it. Explain the likely bias and suggest a correction.

Hints

- Consider what is recorded when a respondent takes no action. - Ask whether one answer requires less effort than the others. - Make every response require the same deliberate action.

Solution

1. A preselected default can influence respondents who overlook the setting or accept it with little consideration. 2. The procedure can systematically increase recorded support for automatic renewal. 3. Present all choices without a preselected answer and require an active selection before submission.

Answer

The default can bias responses toward automatic renewal. Remove the preselection and require respondents to choose among neutrally presented options.

54681912

A survey asks, “How strongly do you disagree that the emergency alerts are not difficult to understand?” Explain the likely source of response bias and rewrite the item so that agreement has a clear meaning.

Hints

- Rewrite the statement without interpreting the response choices first. - Check how many reversals a respondent must mentally process. - Aim for one direct statement whose agreement direction is obvious.

Solution

1. The question combines a negative response direction with a negative statement, making the intended meaning difficult to interpret. 2. Respondents may answer inconsistently because they misread which response represents clear alerts. 3. A clear revision is “The emergency alerts are easy to understand,” with response choices ranging from strongly disagree to strongly agree.

Answer

The double-negative wording can cause response bias through misunderstanding. A clear revision is: “The emergency alerts are easy to understand,” followed by a balanced agreement scale.

54682412

A health survey asks adults how many urgent-care visits they had during the past \(12\) months. Follow-up interviews show that some respondents include memorable visits that occurred \(13\) or \(14\) months ago. Name the data-collection problem and describe its likely effect on the estimated mean number of visits.

Hints

- Compare the stated reference period with the dates respondents actually include. - Decide whether the mistake adds observations or removes them. - Use that direction to judge the effect on the mean.

Solution

1. Respondents are moving events from outside the requested period into the period, a recall problem often called telescoping. 2. Extra visits are counted even though they occurred before the \(12\)-month window. 3. The estimated mean number of visits is therefore likely biased upward.

Answer

This is recall bias caused by telescoping. It will likely bias the estimated mean number of urgent-care visits upward.

54840512

A company emails a survey to all \(2000\) employees. Only \(620\) employees respond, and \(71\%\) of respondents favor a new benefit. An analyst constructs a narrow confidence interval for the proportion of all employees who favor the benefit. a) Identify the main threat to the interval’s validity. b) Explain why surveying the entire employee list does not eliminate the problem. c) What additional assumption would be needed to generalize from respondents to all employees?

Hints

- Distinguish who was contacted from who actually supplied data. - Consider whether opinions could influence the choice to respond. - Identify what must be true about respondents and nonrespondents for the sample proportion to represent everyone.

Solution

1. The main threat is nonresponse bias because most employees did not respond. 2. Sending the survey to everyone does not make the responses representative if the decision to respond is related to opinion about the benefit. 3. Generalization would require assuming that respondents and nonrespondents have similar support proportions, an assumption not justified by the data.

Answer

a) Nonresponse bias. b) The responding group may differ systematically from the employees who did not respond. c) One would need evidence or a credible assumption that response status is unrelated to support for the benefit.

54844012

An online news site posts a poll asking visitors whether they support a subscription fee. From the first \(500\) voluntary responses, the site computes a sample proportion \(\hat p\). An editor plans to use \(E(\hat p)=p\) and \(\sigma_{\hat p}=\sqrt{\frac{p(1-p)}{500}}\) to describe repeated results from polls conducted this way. a) Explain why the editor cannot justify treating \(\hat p\) as an unbiased estimator of the proportion among all site visitors. b) Explain why increasing the number of voluntary responses does not fix the main problem. c) State a sampling method that would support use of the usual sampling-distribution model.

Hints

- Identify who controls whether an observation enters the sample. - Separate random sampling variability from systematic selection effects. - Consider what design would give each population member a known chance to be selected.

Solution

1. Visitors choose whether to respond, so respondents can differ systematically from nonrespondents. The resulting sample proportion need not be centered at the population proportion. 2. A larger voluntary-response sample can reduce random variability around the biased center, but it does not remove the selection bias. 3. A random sample of site visitors, selected from an appropriate sampling frame and invited to respond, would support the usual sampling-distribution model if the other conditions are met.

Answer

a) Voluntary response can create selection bias, so \(E(\hat p)=p\) is not justified. b) A larger sample does not remove systematic bias from self-selection. c) Use a random sample of site visitors from a suitable sampling frame.

54847412

An online retailer receives \(18{,}000\) voluntary responses to a website pop-up question and reports a very narrow \(95\%\) confidence interval for the proportion of all customers who are satisfied. Customers chose whether to answer. a) Explain why the large number of responses can make the reported interval narrow. b) Explain why the interval may still fail to estimate the satisfaction proportion for all customers accurately. c) Distinguish precision from validity in this setting.

Hints

- Consider what aspect of uncertainty sample size addresses. - Identify how people entered the sample and whether that mechanism could be selective. - Separate narrowness of an interval from whether its center targets the correct population value.

Solution

1. A large sample size reduces the estimated standard error, so the formula-based margin of error can be small. 2. Voluntary respondents may differ systematically from customers who ignore the pop-up. Because the sample was not randomly selected from all customers, selection bias can shift the estimate away from the population proportion. 3. Precision refers to small random sampling variability, reflected by a narrow interval. Validity requires a design that supports inference to the target population. A precise interval can be centered at a biased value.

Answer

a) The large sample size reduces the formula-based standard error and margin of error. b) Self-selection can bias the sample, so the interval may be centered away from the true customer proportion. c) The interval can be precise but not valid for the intended population.

54850612

An online news site posts a voluntary poll. A one-proportion test based on \(18{,}000\) responses produces a p-value of \(0.004\). Explain why the small numerical p-value does not by itself justify a conclusion about all readers of the site.

Hints

- Separate the size of the data set from the way participants entered it. - Ask what assumptions make a p-value meaningful for a target population. - Consider whether a small random-error measure can remove systematic selection bias.

Solution

1. A p-value describes sampling variability under the null model only when the data-collection design supports that model. 2. The respondents chose whether to participate, so they may differ systematically from readers who did not respond. 3. A large voluntary-response sample can have very small modeled random error while still having substantial selection bias. 4. Therefore, \(0.004\) does not repair the lack of representative random sampling and cannot support generalization to all readers.

Answer

The p-value addresses random variation under the assumed null model, not voluntary-response bias. Because respondents self-selected, the result cannot be generalized reliably to all readers despite the small p-value.

54855012

An online survey reports that \(41\%\) of respondents support a proposed policy, with a margin of error of \(2.1\) percentage points. The survey link was posted publicly, so people chose for themselves whether to respond, and adults without internet access could not participate. Explain what the reported margin of error measures and why it does not resolve the survey’s main design problems.

Hints

- Separate random sample-to-sample variation from systematic errors in who enters the sample. - Ask which assumptions are required before a margin of error has its usual interpretation. - Consider whether increasing the number of self-selected respondents would fix the source of bias.

Solution

1. A margin of error describes the estimated sampling variability of the sample proportion under the assumptions of the confidence-interval procedure. 2. It does not measure voluntary-response bias or undercoverage because those errors come from how the sample was obtained, not from random sample-to-sample variation. 3. The reported \(2.1\)-percentage-point margin of error therefore cannot make the survey representative of the intended population.

Answer

The margin of error is intended to measure random sampling uncertainty around the sample proportion. It does not account for voluntary-response bias or the undercoverage of adults without internet access, so the reported precision does not make the survey results representative.

54653612

A transit agency posts a survey link only on its social media account and asks riders to rate bus reliability. Identify the main potential sources of sampling bias and explain how they could affect the results.

Hints

- Ask who has a realistic chance to see the survey. - Ask who chooses whether to participate. - Do not claim a direction unless the scenario supports one.

Solution

1. Riders who follow the agency's social media account are more likely to be reached, creating undercoverage of other riders. 2. Participation is voluntary, so riders with unusually strong opinions may be more likely to respond. 3. Either effect can make the respondents unrepresentative of all riders; the direction of the bias cannot be determined from the information given.

Answer

Possible undercoverage and voluntary response bias. Riders not following the account are less likely to be included, and people with strong opinions may be more likely to respond. The direction of the resulting bias is not known.

54654112

A county mails a health survey to a simple random sample of residents. Only about one-fourth return it. Explain the potential bias and identify a difference between respondents and nonrespondents that would matter for estimating the proportion who have a regular primary-care provider.

Hints

- Focus on people who were selected but did not provide data. - Ask what characteristic could affect both responding and the survey variable. - Explain the likely direction only after stating that relationship.

Solution

1. Failure to obtain responses from many selected residents creates potential nonresponse bias. 2. Bias occurs if response behavior is related to whether a resident has a regular primary-care provider. 3. For example, residents who regularly use health services may be more likely to return a health survey, which could make the estimated proportion too high.

Answer

There is potential nonresponse bias. If residents with a regular primary-care provider are more likely to return the survey, the estimated proportion with a provider will tend to be too high.

54655112

To estimate the proportion of county households that own their homes, a researcher uses a printed residential landline directory as the sampling frame. Households without a listed residential landline therefore have no chance to be selected. County records show that households without listed landlines have a lower homeownership rate than households with listed landlines. Identify the likely bias and explain the likely direction of the estimate.

Hints

- Compare the printed-directory sampling frame with the full population of county households. - Identify which households have no chance to be selected. - Use the stated homeownership-rate difference to determine whether omitting that group pushes the estimate up or down.

Solution

1. The target population is all county households, but the sampling frame contains only households with listed residential landlines. 2. Households without listed landlines are omitted, so the design has undercoverage. 3. The problem states that the omitted households have a lower homeownership rate than covered households. 4. The sample therefore overrepresents the higher-homeownership group, so the estimated county household homeownership proportion will tend to be too high.

Answer

Undercoverage bias is likely. Households without listed residential landlines cannot be selected, and the problem states that these omitted households have a lower homeownership rate. Therefore, the estimated proportion of county households that own their homes will tend to be biased upward.

54656112

A parks department estimates average daily trail use by counting visitors only from \(9\) a.m. to \(3\) p.m. on weekdays. Identify the sampling problem and explain why the estimate may not represent all trail users.

Hints

- Compare the stated target period with the times actually observed. - Identify which users cannot appear in the data. - State the direction conditionally unless the scenario guarantees it.

Solution

1. Evening and weekend visitors are excluded from the observation periods. 2. This creates undercoverage because many members of the population have no chance to be counted. 3. If use is heavier outside the selected times, the estimate of average daily use will be too low; the exact direction depends on the unobserved use pattern.

Answer

The plan has undercoverage because it omits evenings and weekends. It cannot represent all trail users and may underestimate daily use if substantial use occurs during the excluded periods.

54656612

A restaurant prints a survey code on receipts with the message, “Tell us what you really think and enter to win dinner for a year.” The manager uses the responses to estimate the mean satisfaction rating of all customers. Explain the potential bias.

Hints

- Identify who controls entry into the sample. - Think about which experiences create stronger motivation to respond. - Avoid assigning a direction when both unusually satisfied and dissatisfied customers may respond.

Solution

1. Customers decide for themselves whether to respond. 2. People with unusually positive or negative experiences may be more motivated to participate. 3. The prize may also attract customers whose response behavior differs from that of typical customers. 4. This voluntary response sample may not represent all customers, and the direction of bias is not determined.

Answer

The survey is subject to voluntary response bias because customers choose whether to participate. Respondents may have stronger opinions or greater interest in the prize than typical customers, so the mean rating may be biased in an unknown direction.

54657612

A school district surveys families about access to after-school programs, but the survey is available only in English. Identify the likely sampling problem and explain how it could affect the estimate of unmet need.

Hints

- Identify which families face a barrier to responding. - Ask whether response likelihood may be related to the survey outcome. - Use that relationship to describe a plausible direction.

Solution

1. Families who cannot comfortably complete an English survey are less likely to respond. 2. This creates potential nonresponse bias because language access is related to participation. 3. If families needing another language face greater barriers to after-school programs, the survey will tend to underestimate unmet need.

Answer

The English-only survey creates potential nonresponse bias because families needing another language are less likely to participate. If those families have less access to programs, the estimated unmet need will likely be too low.

54658112

A city sends a survey to a random sample of residents. After the first mailing, it contacts nonrespondents twice by different methods. Explain how follow-up can reduce nonresponse bias and why it does not guarantee an unbiased estimate.

Hints

- Focus on how the composition of respondents changes after follow-up. - Ask what happens if the hardest people to reach remain different. - Distinguish reducing one bias from eliminating every bias.

Solution

1. Follow-up increases the chance of obtaining data from people who did not answer initially. 2. This can make respondents more similar to the full selected sample. 3. Bias can remain if some people still do not respond and those people differ on the variable of interest. 4. Other biases, such as response bias, may also remain.

Answer

Follow-up can reduce nonresponse bias by bringing more initially missing people into the data set. It does not guarantee an unbiased estimate because remaining nonrespondents may still differ systematically from respondents, and other sources of bias may persist.

54658612

A teacher surveys students who are present on one Monday about the average number of school days missed because of illness. Explain why surveying only students present that day can bias the estimate and state the likely direction.

Hints

- Identify who has no chance to respond on the chosen day. - Connect the reason for exclusion to the variable being estimated. - Use the missing values to determine the direction.

Solution

1. Students absent on the survey day cannot be included. 2. Students who miss more school because of illness are more likely to be absent that day. 3. This undercoverage will tend to omit higher values of the variable. 4. The estimated mean number of illness-related absences will likely be too low.

Answer

The plan undercovers students absent on the survey day, especially students who tend to miss more school. The estimated mean number of illness-related absences will likely be biased downward.

54659612

A transportation survey is conducted only among riders entering the central train station from \(6{:}30\) to \(8{:}30\) a.m. The results are used to estimate how all city residents usually travel. Identify the main bias.

Hints

- Compare the sampled location and time with the entire target population. - Identify transportation groups that cannot be selected. - Determine which travel mode is overrepresented by design.

Solution

1. Only morning train riders at one station can enter the sample. 2. Residents who drive, walk, bike, work different hours, or use other stations are excluded. 3. This convenience sampling frame creates severe undercoverage. 4. Train use will likely be overestimated for all city residents.

Answer

The sample has undercoverage and convenience bias because it includes only morning riders at one train station. It will likely overestimate the proportion of city residents who usually travel by train.

54660612

Before asking residents whether they support a new flood-control fee, a survey shows several dramatic photographs of flood damage. Explain how this survey order can bias responses and propose a fairer procedure.

Hints

- Consider what respondents see immediately before the target question. - Identify the emotion or viewpoint that material encourages. - Redesign the sequence so neither response is favored.

Solution

1. The photographs may prime fear or concern immediately before the policy question. 2. This context can create response bias toward supporting the fee. 3. A fairer procedure presents neutral factual information or asks the support question without persuasive material, using the same neutral wording for all respondents.

Answer

The photographs can prime respondents and bias support upward. Ask the policy question with neutral wording and without emotionally persuasive images, or present balanced information in a standardized way.

54662112

A school board asks for public comments about a boundary change at an evening meeting and treats the speakers' opinions as representative of all district families. Identify the main sampling biases.

Hints

- Separate who can attend from who chooses to speak. - Identify groups that the time and setting may exclude. - Consider which opinions create the strongest motivation to participate.

Solution

1. Families who attend an evening meeting are a convenience group and may differ from families who cannot attend. 2. Only people who choose to speak provide opinions, creating voluntary response bias. 3. Strongly affected or strongly opinionated families may be overrepresented. 4. The comments cannot represent all district families.

Answer

The comments have convenience and voluntary response bias. Meeting attendees are not a random cross-section of district families, and speakers self-select, likely overrepresenting people with strong opinions.

54662612

A wellness survey asks employees to state their current body weight from memory. The analyst then rounds every reported value to the nearest \(10\,\text{lb}\) before calculating the mean. Which step can create systematic response bias, and which step mainly reduces precision without having a guaranteed direction of bias? If employees tend to understate their weight, what direction of bias would that create in the estimated mean?

Hints

- Separate the respondent's reporting behavior from the analyst's rounding rule. - Ask whether each process tends to push values consistently in one direction. - Give a direction for the mean only when the stated behavior supports one.

Solution

1. Reporting weight from memory can create response bias if employees systematically understate or overstate their current weight. 2. Rounding to the nearest \(10\,\text{lb}\) coarsens the measurements, but ordinary nearest-value rounding can move individual values either upward or downward. 3. Therefore, rounding alone has no guaranteed direction of bias. 4. If employees tend to understate their weight, the estimated mean weight will be biased downward.

Answer

Self-reporting from memory can create systematic response bias. Rounding to the nearest \(10\,\text{lb}\) mainly reduces precision and has no guaranteed bias direction by itself. If employees tend to understate their weight, the estimated mean is biased downward.

54663112

A health department wants to estimate the proportion of older county residents who use telehealth. It samples only from the county's online-services account list, which excludes many older residents without reliable internet access. County service records show that older residents without reliable internet access use telehealth less often than those with reliable internet access. Explain the likely bias and direction.

Hints

- Compare who appears on the sampling frame with the full population of older county residents. - Identify the group with no chance to be selected from that frame. - Use the stated telehealth-use difference to determine the direction of bias.

Solution

1. The target population is all older county residents, but the sampling frame is limited to residents on the online-services account list. 2. Many older residents without reliable internet access have no chance to be sampled, creating undercoverage. 3. The problem states that the undercovered group uses telehealth less often. 4. The frame therefore overrepresents the higher-use group, so the estimated telehealth-use proportion will tend to be too high.

Answer

The online-services sampling frame creates undercoverage of older residents without reliable internet access. Because the omitted group has the lower stated telehealth-use rate, the estimated proportion of older county residents who use telehealth will tend to be biased upward.

54663612

A utility company sends a conservation survey only through its mobile app and uses the responses to estimate the proportion of all customers who understand time-of-use pricing. The company defines “understands” as answering at least \(4\) of \(5\) basic pricing questions correctly. In a prior audit, \(74\%\) of app users met this criterion, compared with \(46\%\) of nonusers. Identify the likely bias and direction.

Hints

- Identify which customers have no opportunity to respond through the chosen survey channel. - Compare the stated understanding rates for included and omitted customers. - Decide whether overrepresenting the higher-rate group pushes the estimate up or down.

Solution

1. Customers who do not use the mobile app are excluded from the survey channel, creating undercoverage. 2. The prior audit shows that the undercovered nonusers have a lower rate of meeting the stated understanding criterion than app users. 3. The app-only sample therefore overrepresents customers with the higher understanding rate. 4. The estimated proportion of all customers who understand time-of-use pricing will tend to be too high.

Answer

The app-only survey creates undercoverage of customers who do not use the app. Because the omitted nonusers have the lower stated understanding rate, the estimate of the proportion of all customers who understand time-of-use pricing will tend to be biased upward.

54664612

A randomly selected employee survey is returned by most employees, but many respondents leave the salary question blank. Identify the missing-data problem and explain when it would bias the estimated mean salary.

Hints

- Notice that people returned the survey but skipped one item. - Ask whether missingness is related to the value being estimated. - Use a stated relationship to determine a possible direction.

Solution

1. The blank salary responses are item nonresponse. 2. Bias occurs if employees who omit salary differ systematically in salary from those who report it. 3. For example, if high-salary employees are more likely to omit the item, the reported mean salary will tend to be too low.

Answer

This is item nonresponse. The mean will be biased if willingness to report salary is related to actual salary; for example, greater nonresponse among high earners would bias the mean downward.

54667112

A temperature sensor used in a field study always reads \(1.5\,\text{°C}\) above the true temperature. Explain the effect on estimates of mean temperature and temperature variability.

Hints

- Represent each recorded value as the true value plus a constant. - Consider how adding a constant changes center. - Consider whether distances between observations change.

Solution

1. Every recorded temperature equals the true value plus \(1.5\,\text{°C}\). 2. The estimated mean is therefore biased upward by \(1.5\,\text{°C}\). 3. Adding the same constant to every value does not change differences among values. 4. Measures of variability such as standard deviation are unchanged.

Answer

The mean temperature is biased upward by \(1.5\,\text{°C}\). The variability is unchanged because the same constant is added to every reading.

54667612

Several interviewers translate the same survey question into another language in different ways during live interviews. Explain the potential bias and how to reduce it.

Hints

- Ask whether every respondent hears the same meaning. - Consider how wording differences affect comparability. - Standardize both the translated item and its administration.

Solution

1. Respondents may receive questions with different meanings or tones. 2. Inconsistent wording can systematically influence responses. 3. This creates question wording response bias and reduces comparability. 4. Use a professionally translated, tested, standardized version read the same way to every respondent.

Answer

Inconsistent translation can create response bias because respondents are effectively asked different questions. Use one validated translation and standardized interviewer instructions.

54668112

A manufacturer wants to estimate the proportion of compressors from its 2018 production batch that required a major repair within their first \(5\) years. It surveys only customers whose compressor is still registered in the company's maintenance program. Compressors that failed and were replaced are more likely to have been removed from that program. Identify the main bias and explain the likely direction of the estimated repair proportion.

Hints

- Compare the maintenance-program frame with the full 2018 production batch. - Identify which compressors are especially likely to be missing from the frame. - Use the stated relationship between being omitted and having needed repair to determine the direction of bias.

Solution

1. The sampling frame omits some compressors from the 2018 batch because units removed from the maintenance program cannot be selected. 2. Compressors that failed and were replaced are especially likely to be omitted. 3. Those omitted compressors are also more likely to have required a major repair within \(5\) years. 4. The frame therefore undercovers higher-repair-risk units, so the estimated repair proportion is likely biased downward.

Answer

The design has undercoverage (survivor-related frame bias): failed or replaced compressors are more likely to be missing from the maintenance-program frame. Because those omitted units are more likely to have required a major repair, the estimated \(5\)-year repair proportion is likely biased downward.

54669612

A phone survey asks respondents to choose their most important local issue from six options. Every interviewer reads the options in the same order. Explain a possible response bias and describe a way to reduce it.

Hints

- Think about whether the position of an option could affect how often it is chosen. - Ask what happens when the same options always receive the most noticeable positions. - Look for a redesign that changes order without changing the choices themselves.

Solution

1. Respondents may be more likely to select options heard first because those options receive more attention or are easier to remember during the question. 2. If the same order is always used, this order effect can systematically favor particular choices. 3. The survey can reduce the bias by randomly rotating or randomizing the order of the six options across respondents while keeping the wording unchanged.

Answer

A fixed answer-choice order can create an order effect, systematically favoring options presented in certain positions. Randomizing or rotating the option order across respondents can reduce this response bias.

54670612

A neighborhood survey asks one adult in each sampled household to report how many hours every other household member exercised during the previous week. Identify a likely response problem and explain how it could affect the data.

Hints

- Identify who performs the behavior and who supplies the answer. - Ask whether the respondent can observe the full behavior accurately. - Consider a collection method that gets closer to the person or event being measured.

Solution

1. The respondent may not know the exercise time of every other household member. 2. The answers are proxy reports rather than measurements or self-reports from the people whose behavior is being recorded. 3. Guessing, incomplete observation, or socially desirable assumptions can make the reported values differ systematically from the true values. 4. Collecting responses directly from each household member or using activity records would reduce the problem.

Answer

The survey is vulnerable to proxy-response bias because one person may not accurately know other household members' exercise time. The reports may systematically overstate or understate actual exercise. Direct reports from each person or objective records would be preferable.

54671612

Before asking customers to estimate their checkout wait, a survey states, “Many shoppers wait about \(15\,\text{minutes}\). About how long did you wait?” Explain how the stated value could bias responses and rewrite the question.

Hints

- Consider whether information supplied before the question could influence a person's estimate. - Identify the exact event that should mark the beginning and end of the wait. - Remove information that suggests what a typical answer should be.

Solution

1. The stated \(15\)-minute value can act as an anchor, pulling estimates toward that value even when actual waits differed. 2. The wording may therefore create systematic response bias in the reported wait times. 3. A neutral version is: “About how many minutes did you wait from joining the checkout line until your transaction began?”

Answer

The \(15\)-minute statement can anchor respondents' estimates near \(15\,\text{minutes}\). A neutral item is: “About how many minutes did you wait from joining the checkout line until your transaction began?”

54673112

A survey asks, “Should the city keep the current recycling schedule? Yes or no.” Some residents have no opinion or want a different schedule that is not represented by either response. Explain the potential bias and improve the response options.

Hints

- Compare the available choices with the full range of reasonable opinions. - Ask what a respondent must do when no option fits. - Add choices without steering respondents toward one position.

Solution

1. The forced yes-or-no choices do not represent all plausible opinions. 2. Respondents who are undecided or prefer another schedule must choose an inaccurate answer or skip the item. 3. This can systematically distort the distribution of opinions. 4. The survey should include balanced options such as keep the current schedule, change the schedule, and no preference or unsure, with a follow-up for the preferred change if needed.

Answer

The limited response choices can create response bias by forcing some residents into an inaccurate category. Add balanced options such as “keep it,” “change it,” and “no preference or unsure,” with an optional follow-up about the desired change.

54673612

A survey about a proposed soda tax begins with a large logo and message from an organization campaigning for the tax, although the individual survey questions are neutrally worded. Explain how the survey introduction could still create response bias and suggest a correction.

Hints

- Consider information respondents see before the first question. - Ask whether that information signals a preferred answer. - Separate necessary study information from persuasive material.

Solution

1. The sponsor message reveals the preferred position before respondents answer. 2. Respondents may alter answers to agree with the sponsor, oppose it, or give what they think is socially expected. 3. This can systematically affect responses even when the question wording itself is neutral. 4. Use a neutral introduction that states the study purpose without advocacy branding, and disclose sponsorship separately after responses when ethically appropriate.

Answer

The advocacy message can prime respondents and influence answers toward or against the sponsor's position. Use a neutral introduction without persuasive branding or statements before the survey items.

54675012

A noise monitor records a value only when sound exceeds \(40\,\text{decibels}\); quieter periods are stored as missing. A researcher averages the recorded values to estimate the mean sound level over the entire day. Explain the bias.

Hints

- Identify which values can never appear in the dataset. - Compare the retained values with the full range of the response. - Decide how excluding one end of the distribution affects the mean.

Solution

1. All sound levels at or below \(40\,\text{decibels}\) are excluded from the computed mean. 2. The missingness depends directly on the measured value, so the retained observations are systematically louder than the full set of periods. 3. The average of recorded values will tend to overestimate the true all-day mean sound level. 4. The study needs an instrument that records the full relevant range or a method that properly accounts for values below the detection threshold.

Answer

The monitor systematically omits quieter periods, so the recorded values are not representative of the entire day. Averaging only recorded values biases the estimated mean sound level upward.

54675512

An online course sends a satisfaction survey only to students who complete the final module. The provider uses the results to describe everyone who enrolled. Identify the sampling problem and explain a likely direction of bias.

Hints

- Compare the survey-eligible group with everyone named in the conclusion. - Consider why some students fail to reach the final module. - Decide how those missing students' likely opinions would affect the estimate.

Solution

1. Students who withdrew or stopped participating before the final module cannot be surveyed, so they are undercovered. 2. Completion is likely related to satisfaction and course success. 3. Completers may be more satisfied than noncompleters, so the estimated satisfaction of all enrollees is likely too high.

Answer

The survey undercovers students who did not complete the course. Because noncompleters may be less satisfied, using only completers will likely bias the estimated satisfaction of all enrollees upward.

54676012

A survey asks adults to report the number of minutes they exercised last week. The data contain unusually many responses at \(30\), \(60\), \(90\), and \(120\) minutes. Identify the likely response pattern and explain how it can affect the data.

Hints

- Look for a repeated numerical pattern in the reported values. - Consider how people answer when they do not remember an exact amount. - Think of a collection method that reduces reliance on long-term recall.

Solution

1. The concentration at convenient round numbers suggests heaping caused by recall and rounding. 2. Respondents may replace uncertain values with nearby multiples of \(30\) rather than report exact minutes. 3. The pattern reduces measurement precision and can distort the distribution, especially frequencies near the heaped values. 4. Shorter recall periods or activity logs can reduce the problem.

Answer

The data show heaping: respondents are likely rounding uncertain memories to convenient multiples of \(30\). This response error reduces precision and distorts the distribution. A shorter recall period or recorded activity log would help.

54676512

A salary dataset records every annual salary above \(\$200{,}000\) as exactly \(\$200{,}000\). A researcher uses the recorded values to estimate the mean salary. Explain the likely bias and distinguish it from ordinary rounding.

Hints

- Compare each recorded value above the cutoff with its true value. - Check whether those measurement errors can point in both directions. - Determine how replacing the largest values affects the mean.

Solution

1. Every salary above the cutoff is recorded below or equal to its true value. 2. The errors for high salaries all occur in the same downward direction. 3. The estimated mean salary will therefore tend to be too low. 4. Unlike rounding to the nearest unit, top-coding is asymmetric because it compresses all values above one threshold to the same lower recorded maximum.

Answer

Top-coding biases the estimated mean salary downward because every salary above \(\$200{,}000\) is recorded too low. This is systematic, unlike nearest-unit rounding, which can move values either up or down.

54678012

Before answering a survey item about support for a proposal, each respondent is shown the current percentage of earlier respondents who supported it. Explain how this display could bias later responses and suggest a correction.

Hints

- Ask whether every respondent receives the same information before answering. - Consider how knowledge of a majority position can change a person's response. - Remove information generated by earlier participants from the response process.

Solution

1. The displayed percentage gives respondents information about the apparent majority position. 2. Some respondents may conform to the majority or deliberately react against it, so later answers are influenced by earlier answers. 3. The responses are no longer collected under the same neutral conditions. 4. Hide interim results until data collection is complete and present the item identically to every respondent.

Answer

Showing interim results can create conformity or reactance bias because later respondents are influenced by earlier responses. Do not display results until all responses have been collected.

54678512

During the week after a major holiday, a survey asks shoppers, “How much do you spend in a typical week?” The researcher intends to estimate spending during that specific holiday week. Explain the mismatch and how it could bias responses.

Hints

- Compare the reference period in the question with the period named in the study goal. - Consider whether behavior during the special week differs from ordinary behavior. - Replace the vague time phrase with exact dates.

Solution

1. The phrase “typical week” asks respondents to average over ordinary weeks rather than report the specified holiday week. 2. Holiday spending may be systematically higher or otherwise different from ordinary spending. 3. Responses may therefore underestimate or misrepresent spending during the target week. 4. The item should name the exact dates and ask for spending during that period.

Answer

The question measures ordinary-week spending, not spending during the target holiday week. Because holiday spending may differ systematically, the estimate can be biased. Ask about the exact named dates instead.

54678912

A national online survey opens at 8 a.m. Eastern time and closes as soon as \(2000\) responses are received. Most responses arrive before many people in western time zones begin their day. Explain the likely sampling bias and how to reduce it.

Hints

- Compare the opportunity to respond across time zones. - Identify who is most likely to enter the dataset before the quota is reached. - Replace the response-count stopping rule with a time window and planned selection process.

Solution

1. Early closure makes inclusion depend on when respondents are available relative to Eastern time. 2. People in later time zones have less opportunity to respond before the quota is filled. 3. If time zone is related to the survey variable, the results can be systematically biased. 4. Keep the survey open for a predetermined period covering all time zones and use a probability-based sampling and follow-up plan rather than a first-response quota.

Answer

The early-response quota undercovers people in later time zones and favors those available soon after the Eastern-time launch. Use a fixed response window that covers all time zones and sample respondents with a probability-based method.

54680912

A city survey asks residents to rate a new transit app using these choices: “terrible,” “poor,” “acceptable,” “good,” “very good,” and “excellent.” Explain how the response scale could bias the results, and rewrite the choices as a balanced five-point scale.

Hints

- Compare how many choices express approval with how many express disapproval. - Look for a neutral midpoint and matching categories on both sides of it. - Ask whether the wording gives every overall attitude an equally natural response.

Solution

1. The original scale has more favorable categories than unfavorable categories, which can make positive responses easier or more likely to select. 2. A balanced scale has matching positive and negative categories with a neutral midpoint. 3. One valid revision is “very dissatisfied,” “dissatisfied,” “neither satisfied nor dissatisfied,” “satisfied,” and “very satisfied.”

Answer

The original choices can create response bias because they contain more positive than negative options. A balanced revision is: “very dissatisfied,” “dissatisfied,” “neither satisfied nor dissatisfied,” “satisfied,” and “very satisfied.”

54681412

Researchers place electronic counters on hiking trails to estimate winter use. The counters are more likely to shut down on very cold days, and trail use is usually lower on those days. Explain how deleting all days with missing counts could bias the estimated mean number of hikers per winter day.

Hints

- Identify which days are most likely to be absent from the data. - Compare expected trail use on missing days with use on recorded days. - Decide which direction the remaining observations would pull the mean.

Solution

1. Missingness is related to temperature, and temperature is related to trail use. 2. The deleted days tend to be days with lower trail counts. 3. The remaining data overrepresent warmer, busier days, so the estimated winter daily mean is likely biased upward.

Answer

The estimate is likely biased upward because the missing days are disproportionately very cold days, when trail use tends to be lower.

54661112

An arts organization emails a survey to a random sample of former ticket buyers, but many older email addresses are no longer active. Explain whether the main concern is undercoverage, nonresponse, or both.

Hints

- Distinguish people absent from the sampling frame from people selected but missing a response. - Classify each failure at the point where it occurs. - Explain why either group could differ from the observed respondents.

Solution

1. If inactive addresses remain in the sampling frame and selected people cannot be reached, those selected units become nonrespondents. 2. If the frame itself omits former buyers without usable email addresses, those people are undercovered. 3. Both problems can occur, depending on how the frame and failed deliveries are handled.

Answer

Both may occur. Former buyers omitted from the email frame are undercovered, while selected buyers whose messages fail or who do not reply create nonresponse. Either can bias results if the affected buyers differ from respondents.

54668612

A city randomly selects households for a survey about emergency preparedness. When a selected household does not answer the door, the interviewer immediately surveys the nearest household instead. Explain why this substitution does not solve the nonresponse problem and may introduce additional bias.

Hints

- Track what happens to the originally selected household after it fails to respond. - Ask whether the replacement is chosen by chance from the full sampling frame. - Consider whether neighboring households may resemble one another in ways related to preparedness.

Solution

1. The selected nonresponding households are still missing, so any systematic difference between respondents and nonrespondents remains. 2. The replacement households are chosen by proximity rather than by the original random mechanism. 3. Nearby households may share housing type, income, or neighborhood conditions, so the substitutions can overrepresent characteristics of areas where interviewers already obtained responses.

Answer

The procedure leaves the original nonresponse bias unresolved because the selected households still provide no data. It also adds nonrandom selection: nearest households are convenience replacements and may be systematically similar to the responding neighborhood households.

54670112

A chamber of commerce randomly samples businesses from a directory that has not been updated in three years. Many sampled phone numbers belong to businesses that have closed. Explain the resulting data-collection problem and when it could bias an estimate of current businesses' hiring plans.

Hints

- Compare the directory with the population as it exists now. - Look for both ineligible entries and eligible units that may be absent. - Decide what relationship would make the frame problem affect the hiring estimate.

Solution

1. The outdated frame includes units that are no longer members of the current population, producing many ineligible contacts and missing responses. 2. The problem can also signal that newer businesses are absent from the frame, creating undercoverage of current businesses. 3. The estimate is biased if newly opened or surviving businesses have systematically different hiring plans from the businesses adequately represented by the old directory.

Answer

The outdated directory creates frame errors: closed businesses are included, and newer current businesses may be omitted. Bias results if the current businesses missing from or poorly represented in the frame have different hiring plans from those that can be contacted.

54672112

An interviewer visits sampled households and always asks the person who answers the door for that person's opinion about a city ordinance. The results are reported as opinions of all adults in the sampled households. Identify the selection problem within households and explain a likely source of bias.

Hints

- Separate how households are selected from how a person is selected inside each household. - Ask whether every eligible adult has the same opportunity to respond. - Consider characteristics associated with being home and answering the door.

Solution

1. The adult who answers the door is selected by availability rather than by a random mechanism within the household. 2. Adults who are home at the interview time may differ in age, work schedule, caregiving responsibilities, or opinions from other adults. 3. Therefore, the procedure can systematically underrepresent adults who are less likely to answer the door. 4. The interviewer should randomly select one eligible adult within each sampled household or collect responses from all eligible adults if the design permits.

Answer

The procedure uses nonrandom within-household selection. Adults who answer the door may differ from other adults in ways related to the ordinance, creating bias. Randomly select an eligible adult within each household instead.

54674112

A workplace survey measures morale using ten statements, all phrased so that agreeing indicates high morale. Explain how a tendency to agree with statements could bias the score and describe a design improvement.

Hints

- Consider a respondent who agrees with most statements automatically. - Ask what happens when every agreement receives a high-score interpretation. - Change the item set so the same response pattern does not always push the score one way.

Solution

1. Some respondents tend to agree with statements regardless of their exact content. 2. When every item is keyed in the same direction, this acquiescence tendency can systematically inflate morale scores. 3. The survey can include carefully written statements keyed in both directions, with scoring reversed where appropriate, while avoiding confusing double negatives. 4. Clear neutral wording and validation of the scale can further reduce the problem.

Answer

A general tendency to agree can inflate morale scores because all items point in the same direction. Use a balanced set of clearly worded positive and negative items and reverse-score the appropriate items.

54679412

A phone survey uses automatic speech recognition to record spoken answers. The software fails more often for some accents and marks those responses as missing. Explain the potential bias and suggest a correction.

Hints

- Identify whether recording failures occur equally across respondents. - Consider what other characteristics may be associated with the failure pattern. - Provide a response method that does not depend on the same recognition process.

Solution

1. Missingness depends on speech characteristics rather than occurring randomly. 2. Speakers whose accents are recognized less accurately are underrepresented in the recorded data. 3. If accent is related to geography, language background, or the survey variable, estimates can be systematically biased. 4. Use validated multilingual or accent-robust systems, provide keypad or human-assisted alternatives, and review recognition failure rates across groups.

Answer

The recognition system creates differential missingness and underrepresents speakers whose accents it handles poorly. Offer reliable alternative response modes and validate the system across the population before using the data.

54680212

A health survey interviews younger adults through an anonymous web form but interviews older adults by phone with a live interviewer. The groups report different rates of a sensitive behavior. Explain why survey mode is a possible source of bias.

Hints

- List what differs between the groups besides age. - Consider how privacy and interviewer presence affect sensitive answers. - Redesign the collection so mode does not vary only with age group.

Solution

1. Survey mode is completely linked to age group. 2. Anonymous web responses and live phone responses may produce different levels of privacy, social pressure, and willingness to disclose sensitive behavior. 3. The observed age-group difference may therefore reflect both true age differences and mode-related response bias. 4. Use the same mode across groups or randomly assign modes within each age group so mode effects can be separated from age effects.

Answer

Survey mode is confounded with age group. Live phone interviews may produce different disclosure than anonymous web forms, so the reported difference may partly be a mode effect. Use a common mode or randomize modes within age groups.

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