AP Statistics Quiz: Inference And Experiments

What this quiz covers

This quiz focuses on Inference And Experiments, 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 study shows that sleeping fewer hours causes higher stress for all adults.
2. Because there is a large sample size, the results can be generalized to all employees at the company and establish causation.
3. The study can establish a cause-and-effect relationship between sleep and stress for the volunteers.
4. The study can describe an association between sleep and stress among the volunteers, but it cannot establish causation or be safely generalized to all employees. (correct answer)
5. Because employees self-selected into the study, random assignment is guaranteed, so causation can be concluded.

Explanation: This question evaluates the AP Statistics skill of distinguishing observational studies from experiments and the role of randomization in drawing inferences. The study lacks random assignment, as volunteers self-report sleep and stress without treatment manipulation, making it observational and unable to establish causation due to potential confounders like lifestyle factors. There is no random sampling either, since participants self-selected via email response, so results cannot be safely generalized beyond the volunteers. Choice E distracts by claiming self-selection guarantees random assignment, which is false; random assignment requires researcher control. In a mini-lesson on inference types, observational studies can describe associations in the sample but need random sampling for population generalization and random assignment (turning it into an experiment) for causation. Here, only an association among volunteers can be described, without broader inferences or causal claims.

Question 2

1. Because train stations were randomly selected, the agency can generalize the results to all weekday-morning commuters in the state without concern.
2. Because commuters were not randomly selected within each station (only the first 20 arrivals), selection bias may limit how well the results represent all weekday-morning commuters at those stations. (correct answer)
3. Because the agency observed waiting times, it can conclude that arriving earlier causes shorter waits.
4. Because the total sample size is $15\times 20=300$, the agency can generalize the results to all commuters in the entire country.
5. Because the agency collected data on a Tuesday, the agency can conclude the average waiting time is the same on all days of the week.

Explanation: This question evaluates understanding of complex sampling designs and selection bias. The agency used a two-stage sampling process: random selection of 15 stations (good for generalization) followed by convenience sampling of the first 20 arrivals at each station (problematic). The first 20 commuters to arrive between 7-9 AM may not represent all commuters at those stations - early arrivers might have different travel patterns or waiting experiences than later arrivers. This selection bias within stations limits how well the results represent all weekday morning commuters even at the sampled stations. Additionally, data from a single Tuesday cannot support conclusions about all weekdays. The correct answer identifies the within-station selection bias as the key limitation. Students often focus only on the first stage of sampling and miss bias introduced in subsequent stages.

Question 3

1. Working hybrid causes higher job satisfaction because the company measured all employees.
2. Hybrid work is associated with higher job satisfaction at this company, but the study does not justify a cause-and-effect claim. (correct answer)
3. Hybrid work causes higher job satisfaction because the sample size is large.
4. Hybrid work causes higher job satisfaction because the survey included a comparison group.
5. Hybrid work is associated with higher job satisfaction for all employees nationwide because all employees at this company were surveyed.

Explanation: This question tests recognition of observational studies and their limitations for causal inference. The key skill is understanding that without random assignment, only associations can be established, not causation. The company surveyed all employees about their existing work arrangements and satisfaction levels, making this a census observational study. While the data shows hybrid employees have higher average satisfaction, this association does not prove that hybrid work causes higher satisfaction - there could be confounding factors (perhaps more senior employees choose hybrid work and are also more satisfied). The absence of random assignment to work arrangements means we cannot rule out alternative explanations for the observed difference. Even though all employees were surveyed (a census), this complete enumeration doesn't change the fundamental limitation that observational studies cannot establish causation, only associations.

Question 4

1. The high-fiber snack caused lower afternoon hunger for all adults because matching eliminates all confounding.
2. The high-fiber snack is associated with lower afternoon hunger among gym members, but causation cannot be inferred because the subjects volunteered.
3. The high-fiber snack likely caused lower afternoon hunger for the gym volunteers in this study. (correct answer)
4. Because pairs were matched, the results can be generalized to all gym members in the city.
5. Because the standard snack had the same calories, the study is observational and cannot support a causal conclusion.

Explanation: This question tests understanding of matched-pairs experiments with volunteer subjects. The key skill is recognizing that random assignment within matched pairs allows causal inference, but volunteer samples limit generalization. The researcher created matched pairs based on age and baseline hunger, then randomly assigned treatments within each pair, making this a matched-pairs experiment. The random assignment within pairs controls for the matching variables and allows causal inference - the high-fiber snack likely caused lower afternoon hunger. However, because subjects were volunteers from a gym rather than randomly selected, the causal conclusion applies only to these volunteers, not to all gym members or adults generally. Matching improves precision by controlling for specific variables, but it doesn't enable broader generalization beyond the volunteer sample. The design supports causation for the study participants but not population-wide inference.

Question 5

1. The new routine caused faster 5K times for all runners everywhere because random assignment was used.
2. The new routine caused faster 5K times for runners in the local running club who volunteered for the study. (correct answer)
3. The new routine is associated with faster 5K times, but causation cannot be inferred because the runners volunteered.
4. The new routine caused faster 5K times for all members of the local running club because the runners were randomly assigned.
5. The new routine caused faster 5K times for all adult runners in the city because the study used a control group.

Explanation: This question tests understanding of the scope of causal conclusions in experiments with volunteer samples. The key skill is recognizing that random assignment allows causal inference, but only for the specific group studied when volunteers are used. The sports scientist used random assignment to allocate the 40 volunteers to either the new routine or control group, making this an experiment that can support causal conclusions. However, because the runners volunteered from a local club rather than being randomly selected, the causal conclusion is limited to those volunteers who participated, not all club members or all runners generally. Random assignment eliminates confounding and permits causal inference for the experimental subjects, but without random selection from a larger population, we cannot generalize beyond the study participants. The correct interpretation acknowledges both the causal nature of the conclusion and its limited scope.

Question 6

1. Listening to instrumental music caused higher reading comprehension for first-year students at this university. (correct answer)
2. Listening to instrumental music is associated with higher reading comprehension for first-year students at this university, but causation cannot be inferred.
3. Listening to instrumental music caused higher reading comprehension for all college students because random selection was used.
4. Listening to instrumental music caused higher reading comprehension for the 120 selected students only, but not for other first-year students.
5. Because the music group scored higher, instrumental music will improve comprehension for every individual student.

Explanation: This question tests understanding of experiments that combine random selection with random assignment. The key skill is recognizing that this design allows both causal inference and generalization to the sampled population. The researcher used random selection to choose 120 students from all first-year students, then randomly assigned them to music or silence conditions. This is a true experiment because of the random assignment, which eliminates confounding and allows us to conclude that the music caused higher comprehension scores. Additionally, because the students were randomly selected from all first-year students at the university, we can generalize this causal conclusion to that entire population. The combination of random selection (for generalization) and random assignment (for causation) makes this a powerful design. The causal conclusion applies to first-year students at this university, not just the 120 in the study.

Question 7

1. If Medication A shows a larger mean reduction, the doctor can conclude Medication A caused a larger reduction for patients like those in her clinic. (correct answer)
2. Because patients were randomly assigned, the doctor can generalize the results to all patients with high blood pressure nationwide.
3. Because the doctor used volunteers from her clinic, she can conclude only an association, not causation.
4. Because this is an experiment, the doctor can conclude Medication A causes lower blood pressure for all humans.
5. Because there was no random sampling, the doctor cannot compare the two medications even for her clinic patients.

Explanation: This question probes AP Statistics understanding of experimental inference, emphasizing causation and limits on generalization. Random assignment of patients to medications allows a cause-and-effect conclusion about blood pressure reduction, as it minimizes confounding. However, without random sampling from a broader population, results apply only to similar clinic patients, not nationwide. Choice B is a distractor, incorrectly claiming random assignment enables national generalization. A mini-lesson on inference: experiments with random assignment support causality within the studied group, while random sampling extends to populations; volunteers limit scope. Thus, the doctor can conclude causation for her clinic-like patients.

Question 8

1. Using the mindfulness app caused lower stress for the nurses in the study, and the result can be generalized to all nurses at this hospital. (correct answer)
2. Using the mindfulness app is associated with lower stress for the nurses in the study, but causation cannot be inferred because participation was voluntary.
3. Using the mindfulness app caused lower stress for all nurses nationwide because random selection was used.
4. Using the mindfulness app caused lower stress only for the 150 selected nurses, and the result cannot be generalized even to the hospital's nurses.
5. Because the control group received no app, the study is observational and cannot support a causal claim.

Explanation: This question tests understanding of experiments combining random selection and random assignment for both causation and generalization. The key skill is recognizing when both types of inference are justified by the study design. The researcher first randomly selected 150 nurses from all 900 at the hospital (random selection), then randomly assigned these 150 to app or control groups (random assignment). This design is a true experiment because of random assignment, allowing the conclusion that the app caused lower stress. Additionally, because the 150 nurses were randomly selected from all hospital nurses, this causal conclusion can be generalized to all nurses at the hospital. The combination of random selection from the hospital's nurses and random assignment to treatments creates a powerful design supporting both causal inference and generalization to the specific population sampled (hospital nurses, not all nurses nationwide).

Question 9

1. More sleep causes higher GPA among high school seniors because the sample was randomly selected.
2. There is evidence of an association between reported sleep and GPA among seniors at this school, but a cause-and-effect conclusion is not justified. (correct answer)
3. Because 200 seniors were studied, sleep and GPA must be strongly correlated for all teenagers.
4. Because students self-reported sleep, the researcher can conclude that sleep has no effect on GPA.
5. Because the researcher found a trend, the researcher can conclude that increasing sleep will raise GPA for each individual student.

Explanation: This question assesses understanding of observational studies and correlation versus causation. The researcher used random selection of seniors from the school, which supports generalization to all seniors at that school. However, this is an observational study where students self-reported both sleep and GPA - there was no random assignment to different sleep amounts. The observed association (students reporting more sleep tend to have higher GPAs) could be due to confounding variables like study habits, stress levels, or family support that affect both sleep and grades. Without random assignment, we cannot conclude that sleep causes higher GPA. The correct answer properly identifies an association while rejecting causal claims. A classic error is interpreting any relationship between variables as proof of causation.

Question 10

1. Because participants were randomly assigned, the new package caused higher purchase-intent ratings among these panelists, but generalizing to all consumers is not justified. (correct answer)
2. Because the sample size is 300, the new package caused higher purchase intent for all consumers.
3. Because the participants were not randomly sampled, causation cannot be concluded.
4. Because the study was done online, it is observational and only association can be concluded.
5. Because the new-package group rated higher, the new package is associated with higher intent for all consumers, and causation is justified.

Explanation: This question assesses understanding of experiments with limited generalizability. The study uses random assignment (participants randomly assigned to view different packages), which allows causal conclusions - the new package caused higher purchase intent among these 300 panelists. However, online panelists were not randomly sampled from all consumers, so results cannot be generalized to the broader consumer population. The key distractor is choice C, which incorrectly claims lack of random sampling prevents causal conclusions. In experimental design, random assignment (not random sampling) is what permits causal inference. This study can conclude causation for the participants studied but cannot generalize that causal effect to all consumers due to the convenience sample.

Question 11

1. Because voters were randomly selected, the survey results can be generalized to all registered voters in the county (assuming a high response rate and honest answers). (correct answer)
2. Because the office used a survey, it can conclude that sending text reminders causes higher voter turnout than sending email reminders.
3. Because 1,000 is a large sample, the results can be generalized to voters in every county in the state.
4. Because the voters were randomly selected, the office can conclude that text reminders will increase turnout.
5. Because registered voters were used, the results can be generalized to all adults living in the county, including non-registered adults.

Explanation: This question tests understanding of inference from random sampling in surveys. The election office used random selection from the county's registered voter list, which is the appropriate method for generalizing survey results to that population. With proper random sampling, the survey results can represent the preferences of all registered voters in the county (assuming good response rate and honest responses). However, this is an observational study asking about preferences, not an experiment with random assignment, so no causal conclusions can be drawn about the effect of different reminder types on turnout. The results also cannot extend to non-registered adults or voters in other counties. Students frequently confuse random selection (which supports generalization) with random assignment (which supports causation), or assume large samples automatically allow broader generalization.

Question 12

1. Eating breakfast causes lower BMI because the breakfast group had lower average BMI.
2. There is evidence of an association among the respondents, but the results may be affected by confounding and cannot be generalized to all people because the sample was voluntary and not randomly selected. (correct answer)
3. Because 500 people responded, the blogger can generalize the results to all adults in the country.
4. Because the blogger surveyed followers, the sample is random and supports a causal conclusion.
5. Because BMI was measured, the study is an experiment and supports causation.

Explanation: This question explores limitations of voluntary response surveys in AP Statistics, which are observational and prone to bias. The blogger's social media survey showed an association between breakfast and lower BMI among respondents, but voluntary participation and lack of random assignment prevent causation or generalization due to confounding and bias. Choice A is a distractor, asserting causation from association alone, which overlooks observational design flaws. Mini-lesson on inference types: surveys without random sampling yield biased associations, not generalizable, and observational nature bars causation unlike randomized experiments. Sample size doesn't mitigate volunteer bias. Marked answer B properly highlights association, confounding, and nongeneralizability.

Question 13

1. Because employees were randomly sampled, any difference in mean steps must be caused by the standing desk.
2. Because employees chose the desk type, the study is observational; it may show an association but cannot establish causation, though results may be generalized to the corporation's employees. (correct answer)
3. Because employees were randomly sampled, the results can be generalized to all adults, and causation can be concluded.
4. Because there was no random assignment, the results cannot be generalized even to the corporation's employees.
5. Because step counts were measured with a wearable device, the study is an experiment and can establish causation.

Explanation: This question evaluates AP Statistics distinction between sampling and assignment in inference. Random sampling of employees allows generalization to the corporation's employees, but self-choice of desk type makes it observational, showing only association, not causation, due to potential self-selection bias. Using wearable devices measures outcomes but doesn't create an experiment without assignment. Choice A distracts by confusing sampling with assignment for causation. In a mini-lesson on inference types, random sampling supports population estimates of associations, while random assignment is essential for causality by balancing groups. Here, association in step counts can be generalized corporately but not causally attributed to desks.

Question 14

1. Because classes were randomly assigned to treatments, the district can conclude the platform caused any observed difference in mean test scores. (correct answer)
2. Because classes were randomly assigned, the district can generalize results to all 10th graders in the state.
3. Because this was an experiment, the district cannot draw a cause-and-effect conclusion.
4. Because students were not randomly sampled from all districts, the district cannot draw a cause-and-effect conclusion for these classes.
5. Because there was no random sampling, the district can only conclude an association, not causation.

Explanation: This question assesses the AP Statistics concept of experimental design and its implications for causation and generalization in inference. Here, random assignment of classes to the homework platform or usual system enables a cause-and-effect conclusion because it balances potential confounders across groups. However, there is no random sampling from a larger population; the classes are all 10th-grade algebra classes in the district, so results can be generalized to this district but not necessarily to the state. Choice E is a distractor, wrongly stating that without random sampling, only association can be concluded; in fact, random assignment allows causation even without sampling for broader generalization. A mini-lesson on inference types: experiments with random assignment support causal inferences by creating comparable groups, while random sampling allows extending results to a population; both are ideal but not always necessary together. Therefore, the district can conclude if the platform causes score differences in their classes.

Question 15

1. Because stores were randomly assigned, any observed difference in spending can be attributed to the layout for the 12 volunteer stores. (correct answer)
2. Because stores volunteered, the results can be generalized to all stores in the chain.
3. Because stores were not randomly sampled, no cause-and-effect conclusion is possible even for the 12 stores.
4. Because a coin was used, the results can be generalized to all coffee shops everywhere.
5. Because this is an observational study, only association (not causation) can be concluded.

Explanation: This question addresses AP Statistics experimental design with volunteer subjects and inference scope. Random assignment via coin flips to layouts allows causation for observed spending differences in these 12 stores, controlling confounders. However, since stores volunteered rather than being randomly sampled, results cannot generalize to the entire chain. Choice B is a distractor, wrongly equating volunteering with representativeness for generalization. A mini-lesson on inference: random assignment enables causality in the sample, but random sampling is needed for population extension; volunteers may differ systematically. Thus, causation is justified only for these volunteer stores.

Question 16

1. The results can be generalized to all customers of the chain on Saturdays, but not necessarily to all days of the week. (correct answer)
2. Because stores were randomly selected, the study proves that shopping on Saturday causes customers to spend more.
3. Because every customer at selected stores was included, the results can be generalized to all grocery shoppers everywhere.
4. Because customers were not randomly assigned to stores, no generalization to the chain is possible.
5. Because the sample includes all customers at selected stores, the results can be generalized to all customers of the chain on all days.

Explanation: This question examines AP Statistics inference principles in sampling designs, particularly generalization without causation. Random selection of stores enables generalization to all chain stores, but limiting data to one Saturday means results apply only to Saturdays, not all days, due to potential weekly variations. There is no random assignment or treatment, so this observational census of customers at selected stores cannot prove causation, like Saturday causing more spending. Choice E distracts by overgeneralizing to all days despite the single-day scope. In a mini-lesson on inference types, random sampling supports population estimates but not causality, which requires experimental manipulation; here, it's for estimating Saturday spending across the chain. The design justifies generalization to Saturday customers chain-wide.

Question 17

1. If schools with the requirement have higher graduation rates, the course requirement is proven to cause higher graduation rates.
2. Because data were collected from all schools in the state, the department can conclude the requirement causes higher graduation rates in the state.
3. The department can determine whether there is an association between requiring the course and graduation rate, but it cannot conclude causation. (correct answer)
4. Because the department studied all schools, random assignment is not needed to conclude causation.
5. Because the department used a census of schools, it cannot even examine association.

Explanation: This question tests AP Statistics skills in observational data analysis and inference limitations without randomization. Using existing data from all schools (a census) allows examining associations between course requirement and graduation rates across the state, but without random assignment, causation cannot be concluded due to confounders like school resources. No sampling occurred, but the census covers the population, enabling state-wide association inferences. Choice D distracts by suggesting a census eliminates the need for random assignment for causation, which is untrue. In a mini-lesson on inference types, observational studies (even censuses) show correlations but not causes; experiments with assignment are needed for causality. The department can assess association but not prove the requirement causes higher rates.

Question 18

1. Because patients were randomly selected, the text reminders caused more patients to arrive on time for all patients in the city.
2. Because patients were randomly assigned, the text reminders likely caused a higher on-time rate for the patients in the study, but generalizing to all city patients depends on how well the random sample represents them. (correct answer)
3. Because the study was observational, the higher on-time rate in the reminder group could be due to confounding variables.
4. Because the on-time rate was higher with reminders, the reminders will increase on-time rates in any city, regardless of how subjects were chosen.
5. Because the sample size was 400, the department can conclude the reminders caused higher on-time rates for all patients in the country.

Explanation: This question assesses the skill of determining the scope of inference in a randomized experiment with random sampling, a key concept in AP Statistics inference and experiments. The study involved random selection of 400 patients from the city's appointment list and random assignment to text reminder or no reminder groups, allowing for causal conclusions about the reminders increasing on-time rates for the studied patients and generalization to all patients in the city due to the representative sample. Choice A is a common distractor because it overgeneralizes causation without acknowledging the role of random sampling in supporting population inferences, tempting those who confuse random assignment with random selection. In a mini-lesson on inference types, remember that random assignment supports causal inferences by balancing confounding variables between groups, while random sampling allows generalization from the sample to the broader population. Here, both elements are present, justifying causation within the study and generalization to the city, but not beyond without further evidence. The marked answer B correctly captures this by noting causation in the study and conditional generalization based on sample representativeness.

Question 19

1. Slow-tempo music likely caused higher average spending in the 20 selected stores, but the results may not generalize to all 200 stores because the stores were not randomly sampled. (correct answer)
2. Because stores were randomly assigned, the results can be generalized to all stores in the chain.
3. Because the chain did not randomly assign customers to music conditions, no causal conclusion is possible.
4. Because the 20 stores were close to headquarters, the study is unbiased and supports causation for all stores.
5. Because spending increased, slow-tempo music will increase spending for every customer on every visit.

Explanation: This question tests understanding of experimental inference with nonrandom store selection in AP Statistics, where randomization supports causation but not full generalization. Stores were conveniently selected (closest to headquarters) then randomly assigned to music types, justifying that slow-tempo music caused higher spending in these stores, but convenience sampling limits extension to all 200 stores. Choice B is a distractor, claiming full generalization from random assignment alone, which confuses it with random sampling. Mini-lesson on inference types: random assignment isolates treatment effects for causation, while random sampling ensures representativeness. Bias from selection method restricts scope. Marked answer A correctly balances causation and generalization caveats.

Question 20

1. Adults who support the tax measure are more likely to be registered voters, so the sample is biased and no inference is possible.
2. Because the sample was random, the researcher can generalize the estimated proportion to adults in the state who are on the voter registration list, but the design does not justify a causal claim about what makes adults support the measure. (correct answer)
3. Because the sample size is large, the researcher can conclude the proposed tax measure will increase state revenue.
4. Because the sample was random, the researcher can conclude the tax measure causes adults to support it.
5. Because the researcher surveyed 1200 adults, the results can be generalized to all adults in the United States.

Explanation: This question evaluates understanding of inference from random surveys in AP Statistics, emphasizing generalization to the sampled population without causal claims. The researcher used random sampling from the statewide voter list, supporting estimation of the support proportion for registered voters in the state, but the absence of any treatment or assignment means no causation can be inferred about what influences support. Choice D is a distractor because it wrongly attributes causation to the tax measure itself, misleading those who misinterpret descriptive surveys as experimental. In a mini-lesson on inference types, random sampling enables generalization to the population, but without random assignment, only descriptive or associational inferences are possible, not causal ones. The sample size aids precision but doesn't enable causation or broader generalization beyond the voter list. Marked answer B correctly limits the inference to generalization without causation.