AP Statistics Quiz: Random Sampling And Data Collection

What this quiz covers

This quiz focuses on Random Sampling And Data Collection, giving you a quick way to practice the rules, question types, and explanations that matter most for AP Statistics.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

Question 1

1. The sample is likely representative because selecting every 5th student guarantees randomness.
2. 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. (correct answer)
3. The sample is representative because the sample size (180) is large enough to eliminate bias.
4. The sample is representative because every student had an equal chance of being selected during the two lunches.
5. The sample is not representative only if the researcher randomly assigned students to sleep more or less.

Explanation: 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.

Question 2

1. The sample is likely unrepresentative because it is a voluntary response sample that may overrepresent very satisfied or very dissatisfied customers. (correct answer)
2. The sample is representative because 4,800 is a very large sample size.
3. The sample is representative because every customer had an equal chance to respond by scanning the QR code.
4. The sample is representative because the survey was the same for all respondents, so there is no confounding.
5. The sample is unrepresentative because customers were not randomly assigned to restaurant locations.

Explanation: This question exemplifies voluntary response bias, a major threat to representativeness. When people self-select into a sample (choosing whether to scan the QR code and complete the survey), those with strong opinions are more likely to respond. Very satisfied customers might want to praise the restaurant, while very dissatisfied customers want to complain, but moderately satisfied customers may not bother. This creates a biased sample that doesn't represent the typical customer experience. The large sample size (4,800) cannot fix this fundamental bias - voluntary response samples are inherently unrepresentative regardless of size. True random sampling requires the researcher, not the subjects, to control who is selected.

Question 3

1. The sample is likely not representative because stratified sampling always creates bias by forcing equal numbers from each department.
2. The sample is likely representative because random assignment within departments ensures the results will generalize to all employees.
3. 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. (correct answer)
4. The sample is representative because 300 employees is large enough to be representative of 4,800 employees.
5. The sample is representative only if the company selected the 300 employees by asking for volunteers in each department.

Explanation: 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.

Question 4

1. The sample is likely representative because a simple random sample of all undergraduates was taken and nonresponse was essentially eliminated. (correct answer)
2. The sample is not representative because emailing students is a form of convenience sampling.
3. The sample is representative only if the 250 students were randomly assigned to buy textbooks or not.
4. The sample is representative because 250 is large enough to guarantee it matches the population.
5. The sample is not representative because using a roster means some students had zero chance of selection.

Explanation: In AP Statistics, this question evaluates understanding of sample representativeness through proper random sampling techniques. The method used is simple random sampling from the full roster, with mandatory responses eliminating nonresponse bias. A distractor like choice E wrongly implies that using a roster excludes some students, but actually, labeling and random selection ensure equal chances. Mini-lesson on random sampling: It requires a complete list (sampling frame) and random selection to mirror the population and avoid bias. Here, the SRS and full participation make the sample highly representative for estimating textbook spending.

Question 5

1. The sample is likely to be representative because it is a simple random sample from the full roster, giving each student an equal chance of selection. (correct answer)
2. The sample is representative only if students are randomly assigned to different homework levels.
3. The sample is biased because 80 students is too small to be representative.
4. The sample is biased because random number generators do not create truly random samples.
5. The sample is representative because the principal selected students without looking at their grades, which makes it random assignment.

Explanation: This question tests understanding of simple random sampling, the gold standard for representative sampling. The principal uses a random number generator to select 80 students from the complete school roster, giving every student an equal chance of selection. This is a textbook example of simple random sampling (SRS), which, when properly implemented, produces representative samples. Option A correctly identifies this as a representative sampling method. The distractors contain common misconceptions: sample size of 80 can be adequate depending on the population size and variability, random number generators are valid tools for creating random samples, and this is random selection (not random assignment, which is an experimental design concept). The key principle is that every student had an equal probability of being selected, which SRS achieves. This method should produce unbiased estimates of the mean homework hours for all students in the school.

Question 6

1. The sample is likely representative because it is a stratified random sample with random selection within each department. (correct answer)
2. The sample is not representative because stratified sampling is not random sampling.
3. The sample is representative only if each employee had exactly the same chance of being selected.
4. The sample is representative because 200 employees is a large sample regardless of how they were chosen.
5. The sample is not representative because employees were not randomly assigned to departments.

Explanation: This question tests understanding of stratified random sampling, a valid probability sampling method. The company divided employees into strata (departments) and then randomly selected 50 employees from each stratum. This ensures representation from all departments and can actually improve precision compared to simple random sampling when groups differ. The key is that within each stratum, selection was random using a random number generator. While this gives different selection probabilities if departments have different sizes, stratified sampling is still a representative method. The distractors incorrectly suggest that only simple random sampling is valid or confuse random sampling with random assignment.

Question 7

1. The sample is likely representative because selecting every 20th name is always equivalent to a simple random sample.
2. The sample may not be representative because systematic sampling from an ordered list can be biased if the ordering is related to the variable of interest. (correct answer)
3. The sample is representative because 150 is large enough to ensure representativeness.
4. The sample is representative only if the gym randomly assigned members to join earlier or later.
5. The sample is not representative because it used a random number generator to pick every 20th name.

Explanation: In AP Statistics, this tests systematic sampling's potential biases in data collection. Selecting every 20th from a join-date-ordered list may introduce bias if ordering correlates with childcare interest, like newer members having different family statuses. Distractor A wrongly equates systematic to simple random sampling, but without randomization, patterns can bias. Mini-lesson: Random sampling avoids ordered list pitfalls by ensuring independence; systematic works if no periodicity relates to the variable. Here, the ordering could make the sample unrepresentative for the proportion interested in childcare.

Question 8

1. The sample is likely representative because the chain used random assignment to choose which customers ate on Monday morning.
2. The sample is likely not representative because, although locations were randomly selected, the customers within each location were a convenience sample limited to Monday morning dine-in customers. (correct answer)
3. The sample is representative because the overall sample size is about 1,000, which guarantees representativeness.
4. The sample is representative because selecting 50 locations at random ensures every customer nationwide had an equal chance of being surveyed.
5. The sample is representative because cluster sampling always produces unbiased results regardless of how individuals are chosen within clusters.

Explanation: This question assesses cluster sampling's effectiveness for representativeness in AP Statistics data collection. Locations are randomly clustered, but within-location convenience sampling (Monday mornings) biases toward specific customers. Choice E distracts by claiming cluster sampling always unbiased, but within-cluster selection must be random. Mini-lesson: Random sampling in clusters requires random individual selection inside clusters to mirror the population. The convenience approach here limits representativeness, potentially skewing nationwide satisfaction ratings.

Question 9

1. The 210 respondents form a voluntary response sample, so the results may be biased and not representative of all households. (correct answer)
2. Because the original 600 addresses were randomly selected, the 210 responses are automatically representative of the city.
3. The sample is representative because 210 is a large sample size for a city survey.
4. The sample is representative because random assignment was used to decide who returned the survey.
5. The sample is not representative because simple random sampling can never produce representative samples.

Explanation: This question tests the ability to identify biases in sampling for AP Statistics, particularly in random sampling and data collection. The initial selection is a simple random sample from all addresses, but the final sample is voluntary response due to low return rate, which can bias results toward those motivated to respond. Choice B is a distractor, suggesting the initial randomness carries over despite nonresponse, but nonresponse bias occurs when nonrespondents differ from respondents. A mini-lesson on random sampling: It involves giving each unit an equal, independent chance of selection to ensure representativeness and reduce bias. Voluntary response samples often overrepresent strong opinions, leading to unreliable estimates. Thus, the 210 responses may not represent the city's households.

Question 10

1. The sample is likely representative because surveying at the main trailhead captures the typical visitor.
2. The sample is likely not representative because it undercovers visitors who do not exit at that location and time, so it may be biased. (correct answer)
3. The sample is representative because 120 is a sufficiently large sample size for a park.
4. The sample is representative because the staff randomly assigned visitors to exit between 2 p.m. and 4 p.m.
5. The sample is representative because systematic sampling was used by surveying everyone who exited.

Explanation: Assessing sample representativeness is key in AP Statistics for random sampling and data collection scenarios. This is a convenience sample limited to a specific exit, time, and days, missing visitors using other paths or times. Distractor C claims large sample size ensures representativeness, but size alone doesn't overcome selection bias. Mini-lesson on random sampling: To be representative, every population member must have an equal chance; convenience sampling often biases toward accessible subgroups. In this case, the method undercovers certain hikers, potentially skewing average distance estimates.

Question 11

1. The sample is likely representative because trout were randomly assigned to segments of the river.
2. The sample is likely representative because the segments were randomly selected, making this a cluster sample (by segment). (correct answer)
3. The sample is not representative because weighing every trout in selected segments is not random sampling.
4. The sample is representative because catching trout in the afternoon ensures all trout have an equal chance to be caught.
5. The sample is representative only if at least 30 trout are caught, since large samples are always representative.

Explanation: 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.

Question 12

1. The sample is likely representative because systematic sampling is always equivalent to a simple random sample.
2. The sample is likely representative because 260 is a large enough sample size.
3. The sample may not be representative because it only includes adults who shop at that store on Saturday mornings, which may differ from all adults in the state. (correct answer)
4. The sample is representative because starting with the 3rd adult is random assignment.
5. The sample is representative because every adult in the state had an equal chance to be selected at the grocery store.

Explanation: This question evaluates knowledge of sample representativeness in random sampling and data collection in AP Statistics. The investigators use systematic sampling (every 10th adult) but at a single grocery store on Saturday mornings, creating a convenience sample that may not reflect all state adults, as shoppers there and then might have different habits. For example, weekend shoppers could consume fewer sugary drinks than average. A distractor like choice A claims systematic sampling always equals SRS, but without a random frame covering the entire population, it doesn't ensure representativeness. Mini-lesson on random sampling: Systematic sampling selects every kth individual from a list after a random start, approximating SRS if the list is random-ordered, but it requires a complete population frame to avoid bias; otherwise, it can miss segments of the population.

Question 13

1. The sample is likely not representative because replacing nonresponding households can introduce bias if those households differ systematically (for example, night-shift workers). (correct answer)
2. The sample is representative because stratified random sampling guarantees a perfectly representative sample.
3. The sample is representative because 400 is large enough to overcome any nonresponse problems.
4. The sample is representative only if households were randomly assigned to have longer or shorter commutes.
5. The sample is not representative because random selection within regions is the same as random assignment, which is not allowed for surveys.

Explanation: This AP Statistics question probes nonresponse handling in stratified random sampling for representativeness. The initial stratified random selection is good, but evening-only visits and replacements may bias against households unavailable then, like those with long commutes. Choice B distracts by overclaiming stratified sampling guarantees perfection, but practical issues like timing introduce bias. Mini-lesson: Random sampling within strata aims for balanced representation, but nonresponse adjustments must not create new biases. Replacing based on availability can systematically exclude groups, affecting commute time estimates.

Question 14

1. The sample is likely not representative because random sampling requires random assignment to treatment groups.
2. The sample is likely representative of all residents because 800 addresses were selected, so the sample is large.
3. The sample is likely representative because the returned surveys form a simple random sample of the city.
4. The sample may not be representative because nonresponse could bias results if those who responded differ from those who did not. (correct answer)
5. The sample is representative because every household had an equal chance of responding once the survey was mailed.

Explanation: 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.

Question 15

1. The sample is likely representative because students were randomly selected from the full roster, giving each student an equal chance to be chosen. (correct answer)
2. The sample is not representative because the roster is alphabetized, so the sample will overrepresent certain letters.
3. The sample is representative only if the principal randomly assigns students to join clubs.
4. The sample is likely not representative because 120 is too small to represent a school.
5. The sample is representative because selecting ID numbers is the same as selecting volunteers.

Explanation: 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.

Question 16

1. The sample is likely representative because this is a systematic random sample with a random start, which can be representative if the list has no hidden pattern related to exercise. (correct answer)
2. The sample is not representative because systematic sampling is never random.
3. The sample is representative only if members were randomly assigned to exercise certain days.
4. The sample is representative because selecting every 25th name guarantees equal representation of all workout levels.
5. The sample is not representative because the sample size depends on the length of the list, not on randomness.

Explanation: This question evaluates knowledge of sample representativeness in random sampling and data collection in AP Statistics. The gym uses systematic random sampling by selecting every 25th name after a random start from the full list, which approximates SRS and is likely representative if no periodic pattern in the list correlates with exercise habits. This method is efficient for large lists. A distractor like choice B claims systematic sampling is never random, but with a random start and no patterns, it can be effectively random. Mini-lesson on random sampling: Systematic sampling provides a practical alternative to SRS by evenly spacing selections, ensuring representativeness when the population list is randomly ordered; however, hidden patterns can introduce bias, so it's crucial to verify list randomness.

Question 17

1. The sample is likely representative if the 3 selected departments are typical, but it could be biased if departments differ and only a few clusters were chosen. (correct answer)
2. The sample is representative because surveying everyone in the selected departments eliminates all bias.
3. The sample is representative because departments were randomly assigned to be surveyed.
4. The sample is not representative because cluster sampling cannot be used to make inferences.
5. The sample is representative because the employees in the selected departments form a large sample.

Explanation: This question examines cluster sampling and its potential limitations. The company uses one-stage cluster sampling by randomly selecting 3 departments (clusters) and surveying everyone within them. This method can be representative if departments are relatively similar in satisfaction levels, but could be biased if satisfaction varies substantially between departments and only a few clusters are selected. With only 3 of 12 departments chosen, there's risk of missing important variation. The random selection of departments is crucial - without it, this would be convenience sampling. Cluster sampling trades some precision for practical efficiency, and works best when clusters are internally diverse but similar to each other. The key lesson is that random selection of clusters improves representativeness but doesn't guarantee it, especially with few clusters.

Question 18

1. The sample is likely representative because a simple random sample of accounts was selected from the full subscriber list. (correct answer)
2. The sample is not representative because 2,500 is too small relative to 2,000,000 to represent the population.
3. The sample is not representative because using server logs is a form of voluntary response sampling.
4. The sample is representative only if the company randomly assigned accounts to watch more or less content.
5. The sample is not representative because labeling accounts 1 to 2,000,000 introduces bias.

Explanation: 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.

Question 19

1. The sample is likely representative because the original list was used to randomly select 600 voters.
2. The sample may not be representative because limiting to those who answer weekday afternoon calls can introduce nonresponse/availability bias. (correct answer)
3. The sample is representative because stopping after 250 responses creates a simple random sample of size 250.
4. The sample is representative if the researcher randomly assigns voters to approve or disapprove.
5. The sample is representative because 250 is large enough to overcome any bias from calling times.

Explanation: 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.

Question 20

1. The sample is representative because random-digit dialing ensures every adult in the state has an equal chance to be selected.
2. The sample may be biased because it excludes adults without reachable phone numbers and includes nonresponse from those who do not answer or refuse. (correct answer)
3. The sample is representative because 600 adults is large enough to overcome any bias.
4. The sample is unbiased only if adults are randomly assigned to answer the phone.
5. The sample is unbiased because the agency stopped after exactly 600 interviews, which prevents selection bias.

Explanation: This question addresses coverage bias and nonresponse bias in telephone surveys. Random-digit dialing attempts to create a random sample, but it has inherent limitations. First, it excludes adults without phone access or those with only cell phones if the system targets landlines. Second, it suffers from nonresponse bias because many people don't answer unknown numbers or refuse to participate when they do answer. The 600 who agreed to participate may differ systematically from those who couldn't be reached or refused—perhaps they have more free time, different attitudes toward surveys, or different exercise habits. Option B correctly identifies both sources of bias: exclusion of those without reachable phones (coverage bias) and self-selection among those who could be reached (nonresponse bias). Large sample size alone cannot overcome these systematic biases. True random sampling requires that selection is based on randomization by the researcher, not on availability or willingness to participate.