- A city public health department wanted to estimate the proportion of adult residents who received a flu shot this season. They used a computer to randomly select 800 addresses from the city's list of residential addresses and mailed a survey to each selected household. Of the 800 households, 312 returned a completed survey. Which conclusion is justified based on the design of this study?
Answer: Because only 312 of 800 selected households responded, nonresponse bias may limit how well the results represent all adult residents in the city.
This question assesses understanding of sampling bias and generalization from observational studies. The health department used random selection of addresses, which would normally support generalization to the city's adult residents. However, only 312 of 800 households (39%) responded, creating potential nonresponse bias - those who responded may differ systematically from non-responders in their flu shot behavior. This low response rate severely limits the ability to generalize results to all city residents. The study is observational (no random assignment to get/not get flu shots), so no causal conclusions about flu shot effectiveness can be drawn. Students often mistakenly believe that random selection alone guarantees unbiased results, overlooking the critical impact of nonresponse bias on the validity of generalizations.
- A university wants to determine whether holding review sessions improves performance on a final exam in an introductory statistics course. All 300 students enrolled in the course were invited to participate; 240 agreed. The 240 participating students were randomly assigned to either attend four weekly review sessions or not attend review sessions. The review-session group had a higher mean final exam score. Which conclusion is justified based on the design of this study?
Answer: Attending review sessions likely caused higher final exam scores for the participating students, but generalizing to all students in the course (including nonparticipants) may be limited by nonresponse/volunteer bias.
This question examines inference in experiments with voluntary participation in AP Statistics, focusing on how nonresponse affects generalization. Participating students were randomly assigned to review sessions, supporting causation for higher scores among them, but the 60 nonparticipants introduce volunteer bias, limiting generalization to the full course enrollment. Choice B distracts by overgeneralizing to all university students, disregarding the nonrandom sample. Mini-lesson on inference types: random assignment enables causal conclusions for the studied group, but random sampling or full participation is needed for broader inferences. Large sample size helps but doesn't overcome bias. Marked answer A properly addresses causation and generalization limits.
- A psychologist wants to test whether writing about gratitude reduces stress. She recruited 60 students from her own psychology class and randomly assigned 30 to write about gratitude for 10 minutes each day for two weeks and 30 to write about daily activities for 10 minutes each day for two weeks. After two weeks, the gratitude group had a lower average stress score on a questionnaire. Which conclusion is justified based on the design of this study?
Answer: Writing about gratitude likely caused lower stress for these students, but generalizing to all college students may be limited because the subjects were a convenience sample from one class.
This question evaluates causal inference in experiments using convenience samples in AP Statistics, emphasizing limited generalization. Students from one class were randomly assigned to writing tasks, supporting that gratitude writing caused lower stress for these participants, but the convenience sample restricts broader inferences to all college students. Choice A distracts by incorrectly denying causation due to sampling, when random assignment actually enables it. Mini-lesson on inference types: experiments with random assignment allow causal claims within the group, but without random sampling, generalization is tentative. Control groups strengthen comparisons but don't expand scope. Marked answer B aptly notes causation and sampling limitations.
- A university wants to know whether a brief mindfulness session reduces test anxiety. From all students enrolled in Intro Psychology (about 1200 students), the researchers take a simple random sample of 200 students. These 200 students are then randomly assigned to either (1) attend a 10-minute guided mindfulness session right before an exam or (2) sit quietly for 10 minutes right before the exam. Immediately after the exam, students complete a standardized anxiety questionnaire. Which conclusion is justified based on the design?
Answer: The results can be generalized to all Intro Psychology students at the university, and a cause-and-effect conclusion about mindfulness and anxiety is justified.
This question targets the AP Statistics topic of inference from sampled experiments, focusing on how randomization types enable generalization and causation. Random sampling from Intro Psychology students allows generalization to all such students at the university, while random assignment to mindfulness or quiet sitting supports a cause-and-effect conclusion about anxiety reduction. The results cannot be extended nationwide without sampling from that population. Choice C is a distractor, mistakenly tying causation to random sampling instead of assignment. A mini-lesson on inference: random sampling ensures representativeness for population inferences, random assignment controls variables for causality; together, they allow strong conclusions within the sampled population. Thus, causation is justified for the university's Intro Psychology students.
- A school district wants to determine whether a new online homework platform improves algebra test scores. From all 10th-grade algebra classes in the district, administrators randomly assign half of the classes to use the platform for 6 weeks and the other half to continue with the usual homework system. At the end, all students take the same district test. Which conclusion is justified based on the design?
(Assume the classes included are all 10th-grade algebra classes in the district.)
Answer: Because classes were randomly assigned to treatments, the district can conclude the platform caused any observed difference in mean test scores.
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.
- A city health department wanted to estimate the proportion of adult residents who received a flu shot this year. They used a computer to randomly select 1200 adults from the city's list of registered voters and called them; 864 responded, and 58% of the respondents reported receiving a flu shot. This was an observational study with no treatments assigned. Which conclusion is justified based on the design of this study?
Answer: The sampling method allows generalizing the estimated flu-shot proportion to adult residents in the city, but it does not justify a cause-and-effect claim.
This question tests the distinction between observational studies and experiments regarding inference types. The key skill is recognizing that random selection allows generalization to a population, but observational studies cannot establish causation. The health department used random selection from registered voters, which allows them to generalize their estimate of flu-shot proportion to adult residents in the city (assuming registered voters represent adults adequately). However, since this is an observational study with no random assignment of treatments, we cannot conclude that getting a flu shot causes better health outcomes - there could be confounding variables. The presence of random selection enables population inference (generalization), while the absence of random assignment prevents causal inference. This is a fundamental principle: observational studies can establish associations and make population estimates, but cannot prove causation.
- A researcher wants to test whether listening to music while studying improves memory recall. She recruits 80 students from a dormitory. She pairs students by similar GPA, then within each pair randomly assigns one student to study vocabulary words for 20 minutes with instrumental music and the other to study in silence. Afterward, both take the same recall quiz. Which conclusion is justified based on the design?
Answer: Because students were paired and randomly assigned within pairs, a cause-and-effect conclusion about music and recall is justified for these 80 students.
This question assesses AP Statistics matched-pairs experimental design and its inferential power. Pairing by GPA and random assignment within pairs to music or silence enables a cause-and-effect conclusion about recall for these 80 students, as it controls for ability confounders. Recruiting from a dormitory without random sampling limits generalization beyond these volunteers, not to all college students. Choice D is a distractor, mislabeling it as observational despite assignment. A mini-lesson on inference: matched-pairs experiments with random assignment strengthen causality by reducing variability; sampling is for broader application. Thus, causation is justified for the recruited students.
- A company wanted to know whether allowing employees to work from home two days per week increases job satisfaction. The company surveyed all 300 employees and recorded each employee's current work arrangement (fully in-office vs. hybrid) and their satisfaction score on a 1–10 scale. Hybrid employees had higher average satisfaction. No employees were assigned to a work arrangement by the researchers. Which conclusion is justified based on the design of this study?
Answer: Hybrid work is associated with higher job satisfaction at this company, but the study does not justify a cause-and-effect claim.
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.
- A fitness company wants to test whether a new training app increases the number of push-ups users can do in one minute. The company recruited 80 volunteers who responded to an online advertisement. The volunteers were randomly assigned to use either the new app or the standard app for 6 weeks. After 6 weeks, the new-app group had a higher mean push-up count. Which conclusion is justified based on the design of this study?
Answer: Using the new app likely caused higher push-up counts for these volunteers, but the results may not generalize to all app users because the subjects were not randomly sampled.
This question probes the scope of causal inference in experiments with volunteer samples in AP Statistics, highlighting how lack of random sampling limits generalization. Volunteers were randomly assigned to app groups, supporting that the new app likely caused higher push-up counts for these participants, but the convenience sampling means results may not extend to all potential users due to volunteer bias. Choice B distracts by claiming full generalization based only on random assignment, ignoring that random sampling is needed for population inferences. Mini-lesson on inference types: random assignment allows causation within the study group, while random sampling ensures representativeness for broader generalization. Here, causation is justified, but generalization is cautious. The marked answer A accurately reflects this distinction.
- To test whether background music affects reading comprehension, a researcher randomly selected 120 students from all first-year students at a university. Each selected student was randomly assigned to read a passage either in silence or while listening to instrumental music, then took the same comprehension quiz. The music group scored higher on average. Which conclusion is justified based on the design of this study?
Answer: Listening to instrumental music caused higher reading comprehension for first-year students at this university.
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.
- A sports scientist recruited 40 volunteer runners from a local running club to test whether a new stretching routine reduces 5K race times. The 40 volunteers were randomly assigned to either use the new routine or continue their usual warm-up for 6 weeks, then each ran a timed 5K. The new-routine group had a lower mean time. Which conclusion is justified based on the design of this study?
Answer: The new routine caused faster 5K times for runners in the local running club who volunteered for the study.
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.
- An environmental scientist wants to investigate whether higher daily air pollution is related to more asthma-related emergency room (ER) visits. She collected data for one year in a city, recording the daily average particulate matter (PM2.5) level and the number of asthma ER visits each day. Days with higher PM2.5 tended to have more asthma ER visits. Which conclusion is justified based on the design of this study?
Answer: There is an association between PM2.5 level and asthma ER visits in this city, but causation is not justified because this was an observational study and other variables (like weather or pollen) could confound the relationship.
This question assesses identifying observational studies and their limitations in AP Statistics inference, where no manipulation occurs, preventing causal claims. The scientist collected daily data on PM2.5 and ER visits without assigning treatments, showing an association but not causation due to potential confounders like weather. Choice A is a distractor as it jumps to causation from mere observation over time, tempting those who equate correlation with cause without considering study design. Mini-lesson on inference types: observational studies can establish associations but not causation, unlike experiments with random assignment that control for confounders. Generalization is limited to this city without random sampling. Marked answer C correctly notes the association and confounding risks.
- A marketing firm wants to estimate the mean amount spent per visit by customers at a grocery chain. They randomly select 25 stores from all stores in the chain and then record the total spent by every customer who shops at those stores on a single Saturday. Which conclusion is justified based on the design?
(Assume the 25 stores are selected by simple random sampling.)
Answer: The results can be generalized to all customers of the chain on Saturdays, but not necessarily to all days of the week.
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.
- At a large university, researchers wanted to study whether a brief mindfulness program reduces test anxiety. From all students enrolled in Intro Psychology, 120 volunteers were recruited. The researchers randomly assigned 60 volunteers to attend four 20-minute mindfulness sessions over two weeks and 60 volunteers to receive no sessions. At the end of two weeks, all 120 students completed the same standardized test-anxiety questionnaire. Which conclusion is justified based on the design of this study?
Answer: Because students were randomly assigned, any difference in average test-anxiety scores between groups can be attributed to the mindfulness program for these volunteers.
This question tests understanding of causal inference from randomized experiments. The study uses random assignment of volunteers to treatment (mindfulness) and control (no sessions) groups, which is the key feature that allows causal conclusions. Random assignment ensures that any pre-existing differences between students are distributed equally across groups on average, eliminating confounding variables. However, because participants were volunteers rather than randomly selected from all university students, the results can only be generalized to similar volunteers, not all university students. The correct answer recognizes that random assignment allows attribution of differences to the treatment, but limits this conclusion to "these volunteers." A common misconception is confusing random assignment (which supports causation) with random selection (which supports generalization to a population).
- A researcher completed an experiment to test whether a brief mindfulness exercise reduces test anxiety. At one high school, 10 math classes were available. The researcher randomly assigned 5 entire classes to do a 5-minute mindfulness exercise before a unit test and the other 5 classes to follow the usual pre-test routine. All students then took the same unit test and completed an anxiety scale immediately beforehand; the mindfulness classes reported lower average anxiety. Which conclusion is justified based on the design of this study?
Answer: Because classes were randomly assigned, the mindfulness exercise caused lower reported anxiety for students in these classes, but generalizing to all high school students is not justified.
This question evaluates understanding of cluster randomization in experiments. While individual students weren't randomly assigned, entire classes were randomly assigned to conditions, which still permits causal conclusions - the mindfulness exercise caused lower anxiety for students in these 10 classes. However, classes came from one school (not randomly sampled), so results cannot be generalized beyond this context. The key distractor is choice B, which incorrectly denies causation due to lack of individual randomization. Cluster randomization (randomizing groups rather than individuals) is a valid experimental design that permits causal inference, though it may be less precise than individual randomization. The study can conclude causation for these classes but cannot generalize to all high school students.
- A nutrition researcher wanted to test whether a new high-fiber snack reduces afternoon hunger. She recruited 60 volunteers from a gym and matched them into 30 pairs based on age and baseline hunger rating. Within each pair, one person was randomly assigned to eat the high-fiber snack each afternoon for 2 weeks, and the other was randomly assigned to eat a standard snack with the same calories. At the end of 2 weeks, the high-fiber group reported lower average hunger. Which conclusion is justified based on the design of this study?
Answer: The high-fiber snack likely caused lower afternoon hunger for the gym volunteers in this study.
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.
- A nonprofit completed a field experiment to test whether a new donation-letter message increases donations. From its database of 40,000 prior donors, the nonprofit randomly selected 4,000 donors to include in the study. The 4,000 selected donors were then randomly assigned to receive either the standard letter or the new-message letter. The new-message group donated a higher average amount over the next month. Which conclusion is justified based on the design of this study?
Answer: Because donors were randomly selected from prior donors and randomly assigned, it is reasonable to generalize to all prior donors in the database and to conclude the new message caused higher average donations.
This question assesses understanding of studies with both random sampling and random assignment. Donors were randomly selected from the database of 40,000 prior donors (allowing generalization to that population) AND randomly assigned to receive different letters (allowing causal conclusions). With both design features, we can conclude the new message caused higher donations and generalize this causal effect to all 40,000 prior donors in the database. The key distractor is choice A, which overextends generalization to all people who might receive mail solicitations. In statistical inference, we can only generalize to the population from which we randomly sampled - in this case, the nonprofit's prior donors, not all potential mail recipients. This exemplifies the gold standard: random sampling for generalization plus random assignment for causation.
- A public health team completed an observational study about sleep and blood pressure. They recruited 600 adult volunteers who responded to an online advertisement and recorded each person's average hours of sleep per night and resting systolic blood pressure during one clinic visit. The team found that adults who reported fewer hours of sleep tended to have higher blood pressure. Which conclusion is justified based on the design of this study?
Answer: Adults who sleep fewer hours per night tend to have higher blood pressure in this volunteer group, but causation cannot be concluded.
This question assesses recognition of observational study limitations. The study is observational because researchers simply measured existing sleep habits and blood pressure without any intervention or random assignment. Without random assignment, we cannot conclude that fewer hours of sleep causes higher blood pressure - there could be confounding variables (stress, diet, exercise) affecting both. The key distractor is choice B, which incorrectly claims causation. Additionally, the volunteers responded to an online ad (convenience sample), so results cannot be generalized beyond this volunteer group. In observational studies, we can only identify associations or correlations, never causation, regardless of sample size or how variables are measured.
- A county completed a survey study about screen time and anxiety among its high school students. From a complete roster of all 9,500 high school students in the county, researchers randomly selected 500 students and asked each to report daily screen time and to complete an anxiety questionnaire. Students with higher reported screen time tended to have higher anxiety scores. Which conclusion is justified based on the design of this study?
Answer: Among high school students in the county, higher screen time is associated with higher anxiety scores, but causation cannot be concluded.
This question tests understanding of observational studies with random sampling. Students were randomly selected from all county high school students (allowing generalization to that population), but screen time was not randomly assigned - students reported their existing habits. Without random assignment, this is observational and only association can be concluded. The key distractor is choice A, which incorrectly claims causation. The observed association between higher screen time and higher anxiety could be due to reverse causation (anxious students seek more screen time) or confounding variables (family stress affecting both). Random sampling allows us to generalize the observed association to all high school students in the county, but we cannot conclude screen time causes anxiety.
- To study whether using a standing desk affects daily step counts, a fitness researcher randomly samples 60 employees from a large corporation. However, each sampled employee chooses whether to use a standing desk or keep a traditional desk for the next 4 weeks. Step counts are recorded using the same wearable device for everyone. Which conclusion is justified based on the design?
Answer: 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.
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.
