- A restaurant chain wants to estimate the proportion of all customers nationwide who would recommend the chain to a friend. The chain randomly selects 60 of its 600 locations, then at each selected location surveys the first 10 customers who enter on a Monday morning. Which statement best describes the sample representativeness?
Answer: The sample is likely not representative because surveying only the first customers on Monday morning may not reflect all customers at those locations or nationwide.
This question examines multi-stage sampling with timing bias. While the restaurant locations were randomly selected (good first stage), surveying only the first 10 customers on Monday morning introduces severe bias. Monday morning customers likely differ from evening, weekend, or lunch customers - they might be retirees, shift workers, or business travelers with different dining preferences and loyalty patterns. This temporal bias means the sample doesn't represent all customers at these locations, let alone nationwide. Additionally, 'first 10' is convenience sampling, not random sampling of customers. For a representative sample, the chain should randomly select various days/times and then randomly select customers during those periods.
- A company wants to estimate the proportion of its 4,800 employees who prefer working remotely at least 3 days per week. Employees are divided into departments (Engineering, Sales, Customer Support, HR, Finance). The company randomly selects 60 employees from each department (300 total) using a random number generator and surveys them. Which statement best describes the sample representativeness?
Answer: The sample is likely representative of each department, but it may not represent the overall company well if departments differ in size and equal numbers were taken from each.
This AP Statistics question focuses on stratified random sampling and its impact on representativeness in data collection. The sampling is stratified by department with equal sample sizes, which represents each stratum well but may distort the overall company if departments vary in size. Choice A distracts by claiming stratified sampling always biases, but it's actually useful for ensuring subgroup representation when done proportionally. Mini-lesson: Random sampling within strata helps capture population diversity, but for overall estimates, samples should be proportional to stratum sizes to avoid underrepresenting larger groups. Here, equal samples per department could bias the company-wide proportion if larger departments have different preferences.
- A company wants to estimate the proportion of all employees nationwide who are satisfied with their health benefits. The company divides employees into regions (Northeast, South, Midwest, West) and randomly selects 50 employees from each region to survey, for a total of 200 employees. Which statement best describes the sample representativeness?
Answer: The sample is likely representative because it uses stratified random sampling with random selection within each region.
This question tests comprehension of sample representativeness in random sampling and data collection for AP Statistics. The method is stratified random sampling, dividing the population into regions and randomly selecting within each, which helps ensure representation from diverse groups across the nation. This approach reduces bias by guaranteeing input from each stratum, even if region sizes vary, as estimates can be weighted accordingly. A distractor like choice E argues that equal numbers per region always bias the sample, but this isn't necessarily true if weights are applied or if strata are similar in size; stratified sampling is still a strong method for representativeness. Mini-lesson on random sampling: Stratified random sampling involves dividing the population into homogeneous subgroups and randomly sampling from each, often proportionally, to create a sample that mirrors the population's structure and reduces variability in estimates. This contrasts with simple random sampling, which might underrepresent small groups by chance.
- A school district wants to estimate the average number of hours of sleep per night for all high school students in the district (grades 9–12). A student researcher stands in the cafeteria for two days during lunch and asks every 5th student who walks past a certain hallway entrance to report their typical sleep on school nights, collecting 180 responses. No random number generator was used, and students who did not pass that entrance during those lunches could not be selected. Which statement best describes the sample representativeness?
Answer: The sample is likely not representative because it is a convenience/systematic sample from a specific location and time, which may exclude some groups of students.
This question assesses the skill of evaluating sample representativeness in AP Statistics, focusing on random sampling and data collection methods. The sampling method here is a combination of convenience and systematic sampling, as the researcher only surveys students passing a specific location during limited times without using randomization, potentially excluding students with different lunch schedules or paths. A common distractor is choice A, which incorrectly claims that selecting every 5th student guarantees randomness, but true randomness requires a random starting point and equal chance for all. In a mini-lesson on random sampling, remember that a representative sample gives every individual in the population an equal chance of selection, ideally through simple random sampling using a random number generator. This method minimizes bias and allows results to generalize to the population. Here, the limited access and lack of randomization introduce selection bias, making the sample unrepresentative.
- A school district wants to estimate the proportion of all high school students in the district who get at least 8 hours of sleep on school nights. The population is all students in grades 9–12 across the district's 4 high schools. A researcher visits the district's website, lists every student ID, and uses a random number generator to select 250 IDs; those students are emailed a survey. After two reminders, 138 students respond. Which statement best describes the sample representativeness?
Answer: The 250 selected students form a random sample from the population, but the 138 respondents may be unrepresentative due to nonresponse bias.
This question tests understanding of nonresponse bias in random sampling. While the researcher correctly used random sampling to select 250 students from the population, only 138 responded (55% response rate). The key insight is that the 138 respondents may systematically differ from non-respondents - perhaps students who get adequate sleep are more likely to respond, or conversely, sleep-deprived students might be too busy to answer surveys. Random sampling ensures the initial 250 students form a representative sample, but nonresponse can introduce bias. The correct answer recognizes that while the sampling method was sound, the actual data comes from a potentially biased subset due to nonresponse.
- A city wants to estimate the mean time (in minutes) that all city bus riders wait at stops during weekday mornings. The transit office randomly selects 30 bus stops from a complete list of all stops in the city and posts an employee at each selected stop on a Tuesday from 7:00–9:00 a.m. to record wait times for every rider who arrives during that window. Which statement best describes the sample representativeness?
Answer: The sample is likely representative of weekday morning bus riders in the city because the bus stops were selected using a random method from the full list.
This question assesses understanding of cluster sampling and its representativeness. The transit office used proper random sampling by randomly selecting 30 bus stops from a complete list, making this a cluster sample where each stop is a cluster. Recording all riders at selected stops during the specified time window captures the full variation at those locations. While the sample only represents Tuesday morning riders (not all times/days), it is representative of weekday morning bus riders because the stops were randomly selected. The key principle is that random selection of sampling units (here, bus stops) from a complete frame helps ensure representativeness. This differs from convenience sampling where locations would be chosen for ease rather than randomly.
- A national park wants to estimate the proportion of all visitors this summer who are satisfied with trail signage. The population is all summer visitors. Each day, a ranger flips a coin: if heads, the ranger surveys the first 50 visitors leaving the main trailhead between 4–5 PM; if tails, the ranger surveys the first 50 visitors leaving the visitor center between 4–5 PM. This is done for 20 days, yielding 1,000 surveys. Which statement best describes the sample representativeness?
Answer: The sample is likely unrepresentative because visitors who leave at other times or use other exits have little or no chance of being selected.
This question demonstrates how limited sampling locations and times create bias. While the coin flip adds randomness to location selection, the fundamental problem is sampling only from two exits during one hour (4-5 PM) each day. This systematically excludes visitors who arrive early morning, stay late, use backcountry entrances, or visit on days not sampled. The 4-5 PM timeframe might overrepresent day-hikers and underrepresent campers or serious hikers who return at different times. Additionally, visitor satisfaction might vary by time of day (crowds) or entry point used. Representative sampling requires giving all summer visitors a chance of selection, not just those exiting specific locations at specific times.
- A city wants to estimate the mean amount residents spend on public transportation each month. The population is all adult residents of the city. A student stands outside a downtown subway station from 7–9 AM on Monday and asks every 5th person exiting the station to report their monthly spending; 220 people are surveyed. Which statement best describes the sample representativeness?
Answer: The sample is likely unrepresentative because it systematically excludes adults who do not ride the subway downtown Monday morning.
This question illustrates sampling bias from a poorly chosen sampling frame. The researcher is using systematic sampling (every 5th person), but the critical flaw is sampling only from people exiting a downtown subway station on Monday morning. This systematically excludes many groups: people who work evenings/weekends, those who drive or bike, residents who work from home, retirees, and those living in areas without subway access. The sample will likely overrepresent downtown workers and underrepresent other groups. Remember that representative sampling requires giving all population members a chance of selection - sampling from only one location at one time violates this principle.
- A wildlife biologist wants to estimate the mean weight of all trout in a lake. Over two days, the biologist catches trout using a net near the shore at three easily accessible locations and weighs 75 trout. The biologist uses these weights to estimate the mean weight for all trout in the lake. Which statement best describes the sample representativeness?
Answer: The sample is likely not representative because trout near the shore and at accessible locations may differ from trout elsewhere in the lake.
This question illustrates location bias in ecological sampling. The biologist's method of catching trout only near the shore at easily accessible locations creates a biased sample. Trout behavior, size, and weight often vary by water depth, temperature, and habitat type - shore trout might be younger/smaller or of different species than those in deeper waters. This convenience sampling method doesn't give all trout in the lake an equal chance of capture. For a representative sample, the biologist would need to sample from various depths and locations throughout the lake, perhaps using stratified random sampling based on lake zones. The consistency of using the same net doesn't address this fundamental sampling bias.
- A streaming service wants to estimate the mean number of hours per week that all subscribers watch content. The service randomly selects 2,000 subscriber accounts from its full subscriber database and uses server logs to compute each selected account's viewing hours for the past week. Which statement best describes the sample representativeness?
Answer: The sample is representative because the accounts were randomly selected from the full subscriber list, giving each subscriber an equal chance of selection.
This question demonstrates proper random sampling for estimating population parameters. The streaming service randomly selected 2,000 accounts from their complete subscriber database, giving each subscriber an equal chance of selection - the hallmark of simple random sampling. Using server logs to measure viewing hours is objective and avoids response bias that might occur with self-reported data. The method doesn't require random assignment (that's for experiments, not observational studies) or subscriber participation. This is an excellent example of how technology can facilitate accurate random sampling and data collection. The sample should be representative of all subscribers' viewing habits.
- A wildlife biologist wants to estimate the average weight of trout in a long river. She marks off the river into 20 equal-length segments, randomly selects 4 segments, and then catches and weighs every trout she can net within those chosen segments during one afternoon. Which statement best describes the sample representativeness?
Answer: The sample is likely representative because the segments were randomly selected, making this a cluster sample (by segment).
This question examines sample representativeness in the realm of random sampling and data collection in AP Statistics. The biologist employs cluster sampling by dividing the river into segments (clusters), randomly selecting clusters, and then surveying all trout within them, which can be representative if clusters are similar and randomly chosen. This method is efficient for spread-out populations like a river, where full enumeration is impractical. Choice C is a distractor, claiming it's not random because of censusing within clusters, but the randomness comes from cluster selection, not within-cluster sampling. A mini-lesson on random sampling: Cluster sampling randomly selects groups (clusters) from the population and samples all or some within them, useful for geographic dispersion, but it assumes clusters are internally homogeneous yet similar to each other to maintain representativeness. Unlike stratified sampling, it doesn't ensure representation from every subgroup.
- A streaming service wants to estimate the mean number of hours watched per week by all subscribers. It randomly selects 1,000 subscribers from its database, but then removes any selected subscriber who has not logged in during the past month and replaces them with another randomly selected subscriber, continuing until all 1,000 selected subscribers have logged in recently. Which statement best describes the sample representativeness?
Answer: The sample is likely not representative because excluding inactive subscribers creates undercoverage of low-usage subscribers, biasing the estimate upward.
This question examines sample representativeness in the context of random sampling and data collection in AP Statistics. Although starting with random selection, removing inactive subscribers and replacing them biases the sample toward active users, causing undercoverage of low-usage subscribers and likely overestimating mean watching hours. This alters the sampling frame to exclude part of the population. A distractor like choice A ignores this exclusion, claiming the initial randomness suffices, but post-selection modifications introduce bias. Mini-lesson on random sampling: Random sampling requires maintaining the integrity of the random process throughout; altering the sample after selection, such as excluding subgroups, can introduce bias and undermine representativeness, emphasizing the need for inclusive frames.
- A town wants to estimate the proportion of all registered voters in the town who support a proposed bond measure. The town has an alphabetical list of all registered voters. A pollster starts at the top of the list and surveys the first 500 names. No random numbers are used. Which statement best describes the sample representativeness?
Answer: The sample may be biased because selecting the first 500 alphabetically is not random and could overrepresent certain last-name groups.
This question tests understanding of systematic bias in non-random selection. Taking the first 500 names alphabetically is not random selection and can create bias related to surname patterns. Certain ethnic groups may have last names clustered in specific parts of the alphabet, leading to overrepresentation of some groups and underrepresentation of others. Additionally, some voter registration systems list names by registration date within alphabetical groups, potentially biasing toward longer-term residents. Random sampling requires using a random mechanism (like random numbers) to select individuals, ensuring each person has a known probability of selection. A simple random sample would involve numbering all voters and using random numbers to select 500, eliminating any systematic patterns in selection.
- A county wants to estimate the mean commute time for all employed adults living in the county. A researcher stands outside the county courthouse from 8:00–9:00 a.m. on a Monday and asks 150 people entering the building how long their commute was that day. No random selection is used. Which statement best describes the sample representativeness?
Answer: The sample is likely biased because people going to the courthouse at that time may not represent all employed adults in the county.
This question highlights bias from location-based convenience sampling. People entering the courthouse at 8-9 AM on Monday represent a very specific subset of employed adults - likely those with court business, government employees, or those with flexible schedules. This excludes most workers with fixed morning schedules, those who work evenings/nights, remote workers, and those in locations far from the courthouse. Their commute times may differ substantially from the general population of employed adults. Random sampling would require obtaining a list of all employed adults in the county and randomly selecting individuals to survey. The convenience of standing in one location at one time creates severe limitations on representativeness, demonstrating why random selection from the full population is essential for valid inference.
- A county health department wants to estimate the proportion of all adults in the county who received a flu shot this year. The county is divided into 10 towns. The department randomly selects 2 towns and then surveys every adult who visits the main pharmacy in each selected town during one Saturday. Which statement best describes the sample representativeness?
Answer: The sample is likely not representative because, even with random town selection, surveying only pharmacy visitors on one Saturday can exclude many adults.
This question tests understanding of cluster sampling limitations. While the towns (clusters) were randomly selected, surveying only pharmacy visitors on one Saturday creates multiple sources of bias. Not all adults visit pharmacies, and those who do might be systematically different (perhaps older, more health-conscious, or with chronic conditions requiring medication). Additionally, Saturday visitors might differ from weekday visitors, and people might visit pharmacies in other towns. This method fails to give every adult in the county an equal chance of selection. Proper cluster sampling would require randomly sampling individuals within the selected towns, not just surveying a convenience sample at one location.
- A city wants to estimate the mean time (in minutes) that all commuters in the city spend traveling to work. A statistician stood at a downtown bus terminal for three weekday mornings and asked every 5th person who entered the terminal to report their commute time. The statistician used the collected data to estimate the mean commute time for all city commuters. Which statement best describes the sample representativeness?
Answer: The sample is likely not representative of all city commuters because it includes only people who use that bus terminal during those mornings.
This question examines convenience sampling and its limitations for representativeness. The statistician is using systematic sampling (every 5th person) at one specific location (downtown bus terminal) during specific times (weekday mornings). This method fails to represent all city commuters because it only captures people who use that particular terminal during those times - missing those who drive, bike, walk, use other transit stations, or commute at different hours. The sampling frame (people entering this terminal) doesn't match the target population (all city commuters). For a representative sample of all commuters, the statistician would need to randomly sample from a complete list of commuters or use a method that captures all commuting modes and times.
- A city wants to estimate the mean number of minutes residents spend commuting to work. The city obtains a list of all residential addresses and uses a random number generator to select 800 addresses; a survey is mailed to each selected household and 210 households return completed surveys. Which statement best describes the sample representativeness?
Answer: The sample may not be representative because nonresponse could bias results if those who responded differ from those who did not.
This question evaluates knowledge of sample representativeness related to random sampling and data collection in AP Statistics. The initial selection is a simple random sample from all addresses, but the low response rate introduces nonresponse bias, as those who return surveys might differ from non-respondents in commuting habits. For example, busier commuters might be less likely to respond, skewing the mean. Choice B is a distractor because it claims a large sample size ensures representativeness, ignoring the bias from nonresponse. A mini-lesson on random sampling: Random sampling methods aim to give every population member an equal selection chance, but even with random selection, nonresponse can bias results if responders and non-responders differ on the variable of interest. To mitigate this, follow-up efforts or incentives can improve response rates and enhance representativeness.
- A researcher wants to estimate the proportion of all registered voters in a county who approve of a proposed tax. She uses the county voter list and randomly selects 600 voters. However, she only calls during weekday afternoons, and she records opinions from the first 250 people she reaches who answer the phone. Which statement best describes the sample representativeness?
Answer: The sample may not be representative because limiting to those who answer weekday afternoon calls can introduce nonresponse/availability bias.
This question assesses understanding of sample representativeness in random sampling and data collection for AP Statistics. The initial random selection from the voter list is good, but limiting to weekday afternoon answers introduces nonresponse bias, as those available then might differ in opinions from others. For example, employed voters might be underrepresented. Choice C distracts by saying stopping at 250 creates an SRS, but the availability filter prevents true randomness. A mini-lesson on random sampling: Random sampling aims for equal selection chances, but practical issues like timing can cause nonresponse bias; to counter this, multiple contact attempts or varied times improve response rates and representativeness.
- A streaming service wants to estimate the mean number of hours watched per week among all subscribers. The company labels all subscriber accounts from 1 to 2,000,000 and uses a random number generator to select 2,500 accounts. It then uses server logs (not self-reports) to compute weekly hours watched for those accounts. Which statement best describes the sample representativeness?
Answer: The sample is likely representative because a simple random sample of accounts was selected from the full subscriber list.
This question in AP Statistics examines how simple random sampling ensures representativeness in data collection. The method is SRS from all labeled accounts, with objective data from logs avoiding response bias. Choice B distracts by suggesting sample size is too small relative to population, but representativeness depends on random selection, not absolute size. Mini-lesson: Random sampling uses tools like generators for equal selection probability, allowing inference to the population. Here, the SRS makes the sample likely representative for estimating viewing hours.
- A principal wants to estimate the proportion of students at her school who participate in at least one club. She obtains an alphabetized roster of all students and uses a random number generator to select 120 student ID numbers to survey. Which statement best describes the sample representativeness?
Answer: The sample is likely representative because students were randomly selected from the full roster, giving each student an equal chance to be chosen.
This question tests comprehension of sample representativeness in random sampling and data collection for AP Statistics. The principal uses simple random sampling by randomly selecting ID numbers from the full roster, giving every student an equal chance and likely yielding a representative sample for the school. This method minimizes selection bias effectively. Choice B is a distractor, suggesting alphabetization biases the sample, but random selection from any ordered list still ensures equal probability. A mini-lesson on random sampling: Simple random sampling (SRS) selects individuals such that every subset of size n has equal chance, making it a gold standard for representativeness when a complete list is available, as it avoids systematic exclusion of groups.
