Free math worksheets

- Use the random sample to identify the population. - Determine whether any explanatory treatment was assigned. - Separate estimating a rate from explaining why it occurs.

1. Random selection permits estimation of the defect proportion for all items produced during that shift. 2. The study is descriptive and observational. 3. It can support generalization about the production population but no causal conclusion about why defects occurred.

The sample defect rate may be generalized to estimate the defect proportion among all items produced during that shift. The study does not identify causes of defects.

- Use the sampling method to decide whether population generalization is supported. - Separately ask whether the explanatory condition was imposed by random assignment.

1. The simple random sample supports generalizing a statistically convincing association to the city household population represented by the sampling frame. 2. Solar-panel status was observed rather than randomly assigned. 3. Confounding variables could explain some or all of the association, so a cause-and-effect conclusion is not justified.

The association may be generalized to the city households represented by the random sample. The study does not justify the claim that solar panels cause a change in electricity use because there was no random assignment.

- Use treatment assignment to decide whether a causal comparison is possible. - Use recruitment, separately, to decide how broadly the result can be generalized.

1. The participants were volunteers, so they were not randomly selected from a broader population. 2. The researchers randomly assigned the participating volunteers to the two apps. 3. A statistically convincing difference after random assignment supports a causal conclusion that app assignment affects mean memory score among these participants. 4. Volunteer recruitment does not support statistical generalization of that causal effect to a broader population.

The study can support a causal conclusion that the assigned study app affects mean memory score among the participating volunteers. It cannot statistically generalize that effect to a broader population because the participants were volunteers rather than a random sample.

- Identify the role of the random sample. - Identify the separate role of random assignment. - Combine the two permissions in the final conclusion.

1. Random selection supports generalization to all first-year students at the university. 2. Random assignment supports a cause-and-effect conclusion about the email design. 3. Therefore, the study can estimate a causal effect for the university's first-year student population.

The university may generalize the result to all of its first-year students and conclude that the email design caused the difference in appointment-scheduling rates.

- Use the sampling method to determine the population scope. - Decide whether attendance was imposed as a treatment. - Phrase the relationship as association unless causation is justified.

1. Random selection permits generalization to the population of registered voters in the county. 2. Meeting attendance was observed, not assigned. 3. The result supports an association between attendance and bond support in that population. 4. It does not show that attending a meeting causes support.

The county may generalize the association to all registered voters in the county. It may not conclude that attending a public meeting causes greater support for the bond.

- Identify what random assignment permits. - Examine the narrow age and recruitment setting. - State the most defensible population represented by the participants.

1. Random assignment supports a causal conclusion about the training programs for the participating adults. 2. The statistically significant difference provides evidence for a treatment effect in the experiment. 3. Participants were recruited from two senior centers and were not randomly sampled from a broader older-adult population. 4. Therefore, the causal conclusion should not be statistically generalized beyond the participants studied.

Program A caused a lower mean number of falls among the participants studied. The result should not be statistically generalized to all older adults or to other age groups.

- Identify the population from which boards were randomly selected. - Identify the role of random assignment to inspection procedures. - State the causal effect for the correct production population.

1. Random sampling permits generalization to circuit boards produced that day. 2. Random assignment permits a causal comparison of the inspection procedures. 3. The statistically significant difference supports the conclusion that Procedure B changes the defect-detection rate. 4. The causal conclusion may therefore be generalized to the day's production population represented by the random sample.

The company may conclude that Procedure B causes a higher defect-detection rate and generalize that causal conclusion to all circuit boards produced that day.

- Define the population using both place and time. - Use the sampling frame, not the broader species name, to set the boundary. - Avoid extending the result beyond units that could have been selected.

1. The random sample represents deer in the defined sampling population: the protected area during winter under the observation frame. 2. It does not automatically represent deer in other areas or seasons. 3. The estimate may be generalized only to the population from which the random sample was selected.

The estimate may be generalized to deer in that protected area during the winter sampling period that were included in the observation frame. It should not automatically be generalized to unobserved deer, other areas, other seasons, or other deer populations.

- Identify exactly who could be selected. - Compare the sampling frame with the population named in the proposed conclusion. - Limit generalization to the frame represented by the random sample.

1. The random sample represents the population on the membership list. 2. Residents who are not association members had no chance of selection. 3. Therefore, the result may be generalized to association members but not automatically to all neighborhood residents.

No. The result may be generalized to association members, not to all neighborhood residents, because nonmembers were outside the sampling frame.

- Identify separately how intersections were selected and how signal types were assigned. - Connect one random process to generalization and the other to causation. - State the conclusion using the measured response and the intended population.

1. Random selection of intersections supports generalization to the city's signalized intersections. 2. Random assignment supports a cause-and-effect conclusion about signal type. 3. The statistically significant difference provides evidence that the countdown signal changes the measured response. 4. The city may therefore generalize the causal reduction in the mean number of late-entering pedestrians to its signalized intersections.

The city may conclude that the new countdown signal causes a reduction in the mean number of pedestrians who enter after the warning begins and generalize that conclusion to all signalized intersections in the city.

- Use treatment assignment to determine whether causation is supported. - Define the population from the eligibility rule used before sampling. - Identify the age group that had no chance to enter the study.

1. Random assignment supports a causal conclusion that the scheduling system affects missed appointments. 2. The statistically significant difference provides evidence for that treatment effect. 3. Random selection supports generalization to patients age \(65\) or older represented by the center's sampling frame. 4. Patients younger than \(65\) were outside the frame, so the result should not automatically be generalized to them.

System B causes a lower missed-appointment proportion among patients age \(65\) or older represented by the sampling frame. The study does not establish the same effect for younger patients.

- Use assignment to decide whether the format can be treated as a cause. - Define the population from the roster used for random selection. - Identify educators who were outside that roster.

1. Random assignment supports a causal conclusion about training format. 2. The statistically significant difference provides evidence for the treatment effect. 3. Random selection supports generalization to teachers in the charter-school network represented by the sampling frame. 4. Teachers outside the network had no chance to be selected, so the result does not automatically generalize to all U.S. teachers.

Format A causes higher mean assessment scores among teachers represented by the charter-school network's sampling frame. The study does not automatically generalize to all U.S. teachers.

- Identify the group from which participants were selected by chance. - Identify the condition that was assigned by chance. - Keep the conclusion within the job, organization, and duration actually studied.

1. Random selection supports generalization to the company's current remote proofreaders. 2. Random assignment supports a causal conclusion about using the headphones. 3. The statistically significant difference provides evidence for a reduction in the measured error response during the tested week. 4. The study does not automatically generalize to other occupations, companies, or longer-term use.

For the company's current remote proofreaders, using the noise-canceling headphones caused a lower mean number of proofreading errors during the tested week. The conclusion does not automatically extend to other workers, companies, or longer periods.

- Identify exactly who had a chance to be selected. - Match the target population of an interval to the sampling frame. - Consider how the sampling design would need to change to include the omitted group.

1. The random sample was drawn from the \(5000\) attendees who entered before noon. Also, \(250\le0.10(5000)=500\), and the observed counts are \(145\) and \(105\), both at least \(10\). Therefore, the supported population is all attendees who entered before noon. 2. Later-arriving attendees had no chance to be selected. Their transportation choices could differ, so the sample is not a random sample of all festival attendees. 3. To support inference to all attendees, the organizer should select a random sample from a complete list or sampling frame that covers attendees throughout the festival.

a) All festival attendees who entered before noon. b) Attendees arriving at noon or later were excluded from the sampling frame, so the sample is not representative by design of all attendees. c) Use a random sample from a frame that includes every festival attendee, or a design that randomly samples across the full arrival period.

- Identify whether the data come from a sample or from every member of the stated population. - Decide what source of uncertainty a confidence interval is intended to describe. - Keep the target population limited to the group named in the question.

1. The association observed the entire target population of current members, not a sample from that population. 2. The population proportion is known exactly: \(p=\frac{231}{420}=0.55\). 3. A confidence interval for sampling uncertainty is unnecessary because there is no random sampling error for this finite target population.

No confidence interval is needed for the current membership. The exact proportion is \(0.55\), or \(55\%\), because every current member was included.

- Separate how the observational units were selected from how the explanatory variable arose. - Decide whether the study supports population generalization. - Decide independently whether it supports causation.

1. Random selection permits generalization from the sampled receipts to all purchases made by the chain during that month. 2. Waiting time was observed rather than randomly assigned. 3. The study supports an association but not a cause-and-effect conclusion.

The chain may generalize the observed association to all of its purchases during that month. It cannot conclude that longer waits cause lower satisfaction because there was no random assignment of waiting times.

- Evaluate random assignment and random selection separately. - Link random assignment to the type of relationship that can be concluded. - Limit the population conclusion to the people actually represented by the recruitment process.

1. Random assignment supports a cause-and-effect conclusion about the training plans for the participants. 2. The statistically significant difference provides evidence that the observed treatment contrast is not adequately explained by random assignment variation alone. 3. The participants were volunteers from one fitness center rather than a random sample from a broader population. 4. Therefore, the causal conclusion applies to the participants studied but is not statistically generalizable to all adults.

The study supports the conclusion that the interval-training plan caused greater mean endurance improvement among the participants in this experiment. It does not justify statistically generalizing that effect to all adults because the participants were volunteers from one fitness center.

- Decide whether the data came from a sample or the entire stated population. - Ask whether the explanatory variable was imposed by researchers. - Keep description of a relationship separate from a causal claim.

1. Because data were collected for every player in that league season, the result describes that full population rather than estimating from a sample. 2. Playing time was not randomly assigned as a treatment. 3. The observed relationship is an association and does not establish that additional minutes cause injuries.

The association applies to all players in that league during the season studied because the data form a census of that population. A cause-and-effect conclusion is not justified.

- Examine how participants entered the sample. - Identify the group for which data were actually collected. - Ask whether either variable was imposed as a treatment. - State generalization and causation limits separately.

1. The sample was a convenience sample of willing people at one location. 2. The result cannot be generalized to all city residents, all farmers-market customers, or another defined population. 3. The variables were observed rather than assigned. 4. The study supports an association only among the sixty people interviewed, not a cause-and-effect conclusion.

The study supports an association only among the sixty willing people interviewed at that market. It does not support statistical generalization to a broader population or a claim that buying local food causes greater climate concern.

- Identify the unit that was randomly assigned. - Use the stated statistical evidence when deciding whether the observed treatment difference supports causation. - Examine how the schools entered the study before making a population conclusion.

1. Classes are the units randomly assigned within schools, so the statistically convincing treatment difference supports a causal conclusion for the participating classes. 2. The schools volunteered and were not randomly selected from a larger population. 3. Therefore, the study does not support statistical generalization beyond the participating schools and classes.

The new routine caused higher mean participation scores among the participating classes. The result should not be statistically generalized to all high schools because the schools were volunteers rather than a random sample.

- Use the household selection method to determine the population. - Ask whether solar adoption was imposed by researchers. - Allow for alternative explanations when treatment is self-selected.

1. Random sampling supports generalization to households in the county. 2. Solar adoption was self-selected, not randomly assigned. 3. The study supports an association between solar adoption and electricity bills in the county. 4. It does not establish that installing solar panels caused the full observed difference.

The association may be generalized to county households. A causal conclusion is not justified because households chose whether to install solar panels, so other differences may explain part of the bill difference.

- Use the random sample to identify the population. - Determine whether the exposure was imposed or found in past records. - Account for alternative explanations when stating the relationship.

1. Random sampling supports generalization to adults in the county. 2. The study is retrospective and observational because past exposure was not assigned. 3. The result supports a countywide association. 4. It does not establish that pollution exposure caused the diagnoses because confounding variables may exist.

The association may be generalized to adults in the county. The retrospective observational design does not justify a cause-and-effect conclusion.

- Keep the unit of inference consistent with the unit sampled and assigned. - Use random selection for generalization. - Use random assignment for causation.

1. The work team is the unit sampled and assigned. 2. Random sampling supports generalization to the corporation's domestic work teams represented by the sampling frame. 3. Random assignment supports a cause-and-effect conclusion at the team level. 4. The statistically significant difference supports concluding that Tool B increases mean team output for that population.

The corporation may conclude that Tool B causes higher mean team output and generalize that causal effect to its domestic work teams represented by the random sample.

- Define the population using the record system, place, and time. - Ask whether street features were assigned by researchers. - Notice which events never enter the records.

1. The records form a census of reported crashes in the city during that year. 2. The finding describes that recorded crash population. 3. Street type was not randomly assigned. 4. The study does not establish that the absence of bike lanes caused the higher crash frequency, and unreported crashes are outside the data.

The association describes reported bicycle crashes in that city during the year studied. It does not establish causation and does not automatically include unreported crashes or other years and cities.

- Identify the unit receiving random assignment. - Examine how farms entered the study. - State causation and population scope separately.

1. Plot-level random assignment supports a causal conclusion about irrigation schedule for plots in the participating farms. 2. The statistically significant difference provides evidence for the treatment effect in the experiment. 3. The farms volunteered rather than being randomly sampled from a larger farm population. 4. Therefore, the causal result should not be statistically generalized to all farms.

Schedule A caused higher mean crop yield for plots in the participating farms. The causal result should not be statistically generalized to all farms.

- Separate how the counties entered the study from how the alert systems were chosen. - Ask whether random selection supports generalization. - Ask whether the explanatory condition was imposed by the researchers.

1. The counties were randomly selected, so the observed association may be generalized to counties in the state. 2. The alert systems were not randomly assigned; the agency observed systems already in use. 3. Therefore, the study supports an association between phone alerts and shorter median evacuation times for counties in the state, but it does not establish that phone alerts caused the difference.

The results support a statewide association: counties using phone alerts tend to have shorter median evacuation times. Because the alert systems were not randomly assigned, a cause-and-effect conclusion is not justified.

- Use the selection process to determine the population scope. - Decide what the randomized assignment and stated statistical evidence allow you to say about causation. - Keep the conclusion tied to the exact response and time point that were measured.

1. Random assignment and the statistically convincing difference support a causal conclusion that the charging protocol affects mean remaining capacity after \(100\) cycles. 2. Random selection from the lot supports generalization to all batteries in that production lot. 3. Because the response was measured after \(100\) cycles, the study does not establish which protocol gives greater capacity after substantially more cycles or over the full battery life.

For batteries in this production lot, Protocol B causes a higher mean remaining capacity after \(100\) charge cycles. The result does not establish a long-term advantage beyond the measured \(100\)-cycle period.

- Analyze participant selection separately from treatment assignment. - Connect random selection with the population represented. - Connect random assignment with whether the explanatory condition can be treated as a cause.

1. Study I uses random selection, so its observed association may be generalized to all customers. Because app use is self-selected, it does not establish causation. 2. Study II uses random assignment, so it can establish a causal effect of receiving the app among the participants. Because the participants were volunteers rather than a random sample, it does not support statistical generalization to all customers.

Study I: generalization to all customers is supported; causation is not. Study II: causation is supported for the participants; generalization to all customers is not.

- Define the population from the exact list used for random selection. - Notice which calls never had a chance to appear in that list. - Ask whether the explanatory condition was imposed by the researchers.

1. The sampling frame contains completed customer-support calls in June, so random selection supports generalization to that population of calls. 2. Calls that were abandoned or never completed are outside the sampled population. 3. Interpreter use was observed rather than randomly assigned. 4. Differences in issue complexity, language needs, or customer circumstances could explain the longer times, so causation is not supported.

The association may be generalized to completed customer-support calls in June, not to abandoned or other calls outside that frame. Because interpreter use was not randomly assigned, the study does not show that using an interpreter caused longer resolution times.

- Tie the causal conclusion to the response actually recorded. - Use both the randomized assignment and the stated statistical evidence for that causal conclusion. - Separate a measured intermediate behavior from an unmeasured business outcome.

1. Random assignment and the statistically convincing difference support a causal conclusion about the measured response, email opening. 2. Random selection supports generalization to the retailer's customer population represented by the sampling frame. 3. The study supports that Subject Line B increases the probability of opening the email for that population. 4. Because purchase behavior was not measured, the study does not establish that Subject Line B increases sales or purchase rates.

Subject Line B causes a higher email-open rate for the customer population represented by the random sample. No conclusion about purchases or sales is supported because those outcomes were not measured.

- Define the population using the time period in the sampling frame. - Determine whether the explanatory variable was assigned or merely recorded. - Consider whether a later system change makes the old population comparable to the new one.

1. Random selection supports generalization to the 2025 user sessions represented by the sampling frame. 2. Session entry method was observed rather than randomly assigned, so only an association is supported. 3. The redesign may have changed user behavior and the meaning of the entry categories. 4. Therefore, the result should not automatically be generalized to 2026 sessions under the redesigned system.

The study supports an association between entry method and session duration for the 2025 sessions represented by the sample. It does not establish causation and should not automatically be generalized to sessions after the 2026 redesign.

- Determine whether the study used all units in a defined finite population. - Combine the random assignment with the stated statistical evidence for causation. - Identify broader groups that were not represented through random selection.

1. Random assignment and the statistically convincing difference support a causal conclusion that System B reduces the mean number of order errors for the \(12\) participating branches under the study conditions. 2. Because all current branches were included, no sampling inference is needed for that finite set. 3. The branches were not randomly sampled from a broader population of possible or future branches. 4. The result should not automatically be generalized to future branches, branches in other regions, or substantially different operating conditions.

System B causes a lower mean number of order errors for the company's \(12\) current branches under the study conditions. The study does not by itself support generalization to future branches or branches outside this regional company.

- Check whether the policy was assigned by a random mechanism. - Check whether the states were sampled from a larger population. - Keep the conclusion limited to the states and period actually observed.

1. The policy condition was not randomly assigned, so differences between the states provide alternative explanations for the crash-rate change. 2. The study does not establish that the law caused the larger decrease. 3. Because only these two deliberately chosen states were studied, the result does not support broad statistical generalization to other states. 4. The appropriate conclusion is a descriptive association for the two states during the studied period.

The study shows that the adopting state had a larger crash-rate decrease than the comparison state during the studied period. It does not establish causation and does not support generalization to other states.

- Use the sampling process to set the population scope. - Ask whether researchers imposed the explanatory condition. - Phrase the conclusion as a relationship rather than an intervention effect.

1. Random selection supports generalizing the association to adult patients in the health-system population represented by the sampling frame. 2. The genetic marker was observed rather than assigned by researchers. 3. The study supports an association between the marker and condition prevalence. 4. It does not show that changing or removing the marker would cause a change in the condition, and other biological factors may explain the relationship.

Among adult patients represented by the sampling frame, the genetic marker is associated with higher prevalence of the condition. The observational study does not establish a cause-and-effect relationship.

- Use treatment assignment and the stated statistical evidence to decide whether causation is supported. - Define the population from the exact frame used for random selection. - Identify customers who were excluded before sampling began.

1. Random assignment and the statistically convincing payment-rate difference support a causal conclusion that Version B increases on-time payment. 2. Random selection supports generalization to customers in the English-language sampling frame. 3. Customers using other communication languages had no chance to be selected. 4. Therefore, the result should not automatically be generalized to the retailer's entire customer population.

Version B causes a higher on-time payment rate among customers represented by the English-language sampling frame. The result does not automatically generalize to customers who use other communication languages.

- Tie the causal statement to the environment in which the response was measured. - Use the random sample to identify the employee population represented. - Use the stated statistical evidence only for simulated performance, not for unmeasured real-emergency behavior.

1. Random assignment and the statistically convincing difference support a causal conclusion about performance in the computer simulation. 2. Random selection supports generalization to the employee population represented by the sampling frame. 3. The measured response is simulated evacuation time, not behavior during an actual emergency. 4. Therefore, Format B can be said to reduce simulated evacuation time for that employee population, but the study does not establish the same effect in a real emergency.

Format B causes shorter evacuation time in the computer simulation for employees represented by the random sample. The result does not by itself establish that Format B would shorten evacuation time during an actual emergency.

- Tie the causal statement to the exact survey response. - Use the random sample to identify the voter population represented. - Separate what respondents say they plan to do from what they later do.

1. Random assignment supports a causal conclusion that the description changes stated voting intention. 2. The statistically significant difference provides evidence for that treatment effect. 3. Random selection supports generalization to the voter population represented by the sampling frame. 4. The measured response is stated intention, not an actual vote cast later. 5. Therefore, the study does not establish an effect on actual voting behavior.

For voters represented by the random sample, the description causes a change in stated voting intention. The study does not show that it changes actual votes because actual voting behavior was not measured.

- Separate what random selection permits from what random assignment permits. - Identify exactly which packets were eligible to enter the study. - Notice the cultivar and environmental conditions that were held fixed.

1. Random assignment supports a cause-and-effect conclusion about the coating. 2. The statistically significant difference provides evidence for that treatment effect. 3. Random selection from the current inventory supports generalization to packets of this cultivar in that inventory. 4. The study does not justify generalization to other cultivars or substantially different growing conditions.

The new coating caused a higher germination rate for packets of this tomato cultivar in the company's current inventory under the tested greenhouse conditions. The result cannot automatically be generalized to other cultivars or growing conditions.

- Decide whether the study used a sample or included the entire stated group. - Ask whether the explanatory condition was assigned by chance. - Keep the conclusion within the location and time period actually observed.

1. Because every 2025 call at the airport was included, the observed association describes that finite set of calls without sampling uncertainty. 2. Staffing was not randomly assigned, so other factors such as crowd size or incident severity may be related to both staffing and response time. 3. The study does not establish causation and does not automatically generalize to other airports or years.

For this airport’s 2025 calls, higher staffing was associated with shorter response times. The study cannot show that higher staffing caused the shorter times, and it does not automatically support conclusions about other airports or years.

- Identify the exact sampling frame. - Decide whether the thermostat setting was imposed or merely observed. - Keep the conclusion within the technology and time period represented in the data.

1. Random selection supports generalization to the utility’s customer homes with smart thermostats. 2. Thermostat settings were observed rather than randomly assigned, so the study supports association but not causation. 3. The conclusion does not automatically extend to homes without smart thermostats or to seasons other than July 2026.

Among the utility’s customer homes with smart thermostats, lower daytime thermostat settings were associated with greater electricity use during July 2026. The study does not establish causation and does not automatically apply to homes without smart thermostats or to other seasons.

- Separate the role of random selection from the absence of random assignment. - Look for pool characteristics that could affect both staffing and rescues. - Avoid interpreting the direction of an association as the direction of a treatment effect.

1. Random selection supports generalizing the observed association to public swimming pools in the state during the summer studied. 2. Lifeguard staffing was not randomly assigned, so the study does not support causation. 3. Pools with stronger currents, larger crowds, or higher-risk activities may both employ more lifeguards and require more rescues, reversing the naive causal interpretation.

The association can be generalized to the state’s public swimming pools during the summer studied. It does not show that adding lifeguards causes more rescues, because higher-risk pools may both schedule more lifeguards and have more rescues.

- Identify the response variable actually recorded in the experiment. - Use the two kinds of randomization to determine population and causation. - Do not replace the measured technical outcome with an unmeasured customer outcome.

1. Random selection supports generalization to the retailer's current website pages under the standardized test conditions. 2. Random assignment supports the conclusion that image format affects page-load time. 3. The statistically significant difference provides evidence for the reduction in the measured technical response. 4. Customer behavior was not measured, so the experiment does not show that the format increases purchases, customer satisfaction, or retention.

For the retailer's current website pages under the standardized network test conditions, the new format caused faster mean page-load time. The study does not show that the format increases purchases, customer satisfaction, or retention.

- Decide whether chance determined who received the program opportunity. - Use the stated statistical evidence for the comparison among lottery entrants. - Distinguish lottery entrants from the broader set of all eligible households.

1. The lottery assigns the opportunity to receive a voucher by chance, making the winner and nonwinner groups comparable for causal inference. 2. The statistically convincing difference supports a causal conclusion about being offered a voucher among people who entered this lottery. 3. The applicants were not randomly selected from all eligible households, so the result does not automatically generalize to eligible households that did not apply or to other cities and program designs.

For people who entered this city's lottery, being offered a housing voucher caused a higher probability of remaining continuously housed for the following year. The result does not automatically generalize to nonapplicants, other cities, or different voucher programs.

- Trace the study in chronological order: frame, sample, then treatment assignment. - Compare the roster population with the population named in the investigative question. - Use sampling for population scope and assignment for causation, then apply the frame limitation.

1. The simple random sample of \(200\) students from the active-email roster is the random-sampling step. 2. Randomly assigning sampled students to Video A or Video B is the random-assignment step. 3. Students with inactive email accounts are absent from the sampling frame, creating undercoverage relative to all district high school students. 4. Random assignment and the statistically convincing score difference support a causal conclusion that Video B raises mean quiz score for students represented by the study. 5. Random sampling supports generalization to the active-email-roster population, but the undercoverage prevents an automatic generalization to all district high school students.

a) Random sampling: an SRS of \(200\) students from the active-email roster. Random assignment: sampled students are assigned by chance to Video A or Video B. b) Undercoverage: students with inactive school email accounts are missing from the sampling frame. c) Video B causes a higher mean algebra quiz score for students represented by the active-email roster. The result can be generalized to that frame population, but not automatically to all district high school students.

- State the outcome exactly as it was measured. - Combine the randomized assignment with the stated statistical evidence when deciding whether causation is supported. - Keep measurement limitations separate from population limitations.

1. Random assignment together with the statistically convincing treatment difference supports a causal conclusion about the effect of app assignment on self-reported stress for the participants. 2. The response is self-reported, so reporting behavior may not equal physiological stress. 3. The study supports an effect on reported stress, not automatically on every objective measure of stress. 4. Because the participants were volunteers rather than a random sample, statistical generalization to all adults is not justified.

The app caused lower mean self-reported stress among the participants. The study does not necessarily show the same effect on objective stress measures and should not be statistically generalized to all adults.

- Distinguish the condition assigned by the researchers from the behavior later chosen by participants. - Use the randomized offer and the stated statistical evidence for the causal statement. - Keep the population conclusion limited by how participants entered the study.

1. Random assignment supports a causal conclusion about the treatment actually assigned: being offered a free transit pass. 2. The statistically convincing difference shows that the offer affected missed appointments among the participants. 3. Because some assigned patients did not use the pass, actual pass use was not randomly assigned. 4. The study therefore supports a causal effect of offering the pass, not a causal effect of actually using the pass or riding transit; volunteer recruitment also prevents statistical generalization beyond the participants.

The study supports that offering a free transit pass caused fewer missed appointments among the participants. It does not establish that actually using the pass or riding transit caused the reduction, and it should not be statistically generalized beyond the participants.

- Use random selection and random assignment for different parts of the conclusion. - Account for the stated statistical evidence before making the causal statement. - Note characteristics of the setting, duration, and population that were not varied or sampled.

1. Random assignment together with the statistically convincing treatment difference supports a causal conclusion that the tested warmer condition reduces mean \(30\)-day mass gain. 2. Random selection supports generalization to juvenile trout from the sampled lake. 3. The conclusion is limited to the two tested temperature conditions and the \(30\)-day response period. 4. The experiment does not automatically establish the same effect for other lakes, adult trout, or natural lake conditions that differ from the laboratory.

For juvenile trout from this lake under the tested laboratory conditions, the warmer temperature causes lower mean mass gain over \(30\,\text{days}\). The result is limited to the tested temperatures and time period and should not automatically be generalized to other trout populations or field conditions.

- Identify exactly which treatment conditions were randomly compared. - Use the stated statistical evidence only for contrasts that the randomized design actually contains. - Notice the comparison condition that is absent.

1. Random assignment and the statistically convincing difference support a causal comparison between the two tested doses among the participants. 2. The study supports that the high dose causes a larger mean improvement than the low dose among those participants. 3. Because no group received placebo or no treatment, the study cannot determine whether either dose improves reaction time compared with receiving no supplement. 4. Volunteer recruitment prevents statistical generalization beyond the participants.

The high dose causes a larger mean improvement than the low dose among the participants. The study does not show that either dose is better than no supplement because no placebo or no-treatment group was included, and it should not be statistically generalized beyond the participants.

- Identify the observational unit represented by each data point. - Keep the conclusion at the same level as the measured variables. - Ask whether any individual person's exposure and outcome were paired in the data.

1. Random selection of counties supports generalizing the observed county-level association to counties in the population represented by the sample. 2. The variables are county averages or rates, so the observational units are counties. 3. The study does not link individual residents' park access with their individual obesity status. 4. Therefore, it cannot conclude that a particular person with greater park access is less likely to have obesity, and it does not establish causation.

The study supports a population-level association among counties: counties with more park acreage per resident tend to have lower adult obesity rates. It does not support an individual-level claim about residents, and it does not establish that parks cause lower obesity rates.

- Identify the smallest unit that receives its own treatment assignment. - Use the stated statistical evidence with the building-level randomization. - Phrase the causal conclusion at the same level as the randomized treatment.

1. Buildings are the experimental units because the composting program is assigned to whole buildings. 2. Random assignment plus the statistically convincing participation difference supports a causal conclusion about offering the building-wide program. 3. Random selection of buildings supports generalization to apartment buildings in the city represented by the sampling frame. 4. The conclusion concerns the effect of a building-level program on resident participation, not separate independent treatment assignments to individual residents.

For apartment buildings in the city's sampling frame, assigning a building-wide composting program causes a higher mean resident compost-participation rate. The treatment effect is a building-level program effect, not an independently randomized individual-resident effect.

- Follow the sample from initial selection through actual participation. - Identify the stage at which self-selection enters. - Use the stated statistical evidence with random assignment for causation, but use the final participating group for population scope.

1. Random assignment and the statistically convincing score difference support a causal conclusion among the students who participated. 2. Although the invitation list was randomly selected, participation depended on consent. 3. If consenters differ from nonconsenters in ways related to the response, the participating group is not a random sample of all students. 4. The result supports that Tool A causes higher mean quiz scores for the participating students, but it does not support statistical generalization to the full university enrollment.

Tool A causes higher mean quiz scores among the students who participated. Random assignment supports causation, but voluntary consent prevents statistical generalization to all university students.

- Use within-pair random assignment to decide whether causation is supported. - Identify what the pairing controls. - Use the recruitment method, not the assignment method, to set the population scope.

1. Random assignment within twin pairs supports a causal conclusion about diet for the participating twins. 2. The statistically significant difference provides evidence for the treatment contrast. 3. Matching twins controls genetic and many shared-background differences. 4. The pairs volunteered through a specialized registry rather than being randomly selected from all adults. 5. Therefore, the causal conclusion should not be statistically generalized beyond the participating twins.

The high-fiber diet caused lower mean cholesterol after six weeks for the participating twins. The matched-pairs randomization supports causation, but volunteer recruitment from a twin registry prevents statistical generalization to all adults.

- Identify the condition required for a patient to enter the dataset. - Keep the conclusion within the group satisfying that condition. - Ask whether the excluded patients are necessary for studying the survival outcome.

1. The dataset is restricted to patients who survived at least \(30\,\text{days}\), so conclusions apply only to that survivor population. 2. Rehabilitation timing was observed rather than randomly assigned, so the study supports an association, not causation. 3. Patients who died within \(30\,\text{days}\) are excluded by design. 4. Because the outcome used to define inclusion removes all nonsurvivors, the study cannot estimate or compare \(30\)-day survival for rehabilitation groups.

Among \(30\)-day survivors, earlier rehabilitation is associated with shorter hospital stays. The study does not establish causation and cannot draw conclusions about \(30\)-day survival because nonsurvivors were excluded.

- Identify the smallest unit that receives its own promotion assignment. - Combine the randomized assignment with the stated statistical evidence for the causal conclusion. - Use the sampling frame to determine the population of units represented.

1. The experimental unit is a store-day because the promotion is assigned separately to each selected store on a particular day. 2. Random assignment and the statistically convincing sales difference support a causal conclusion that the promotion increases mean daily sales. 3. Random selection of store-days supports generalization to the October–December 2026 population of store-days represented by the schedule. 4. The study does not provide independently randomized treatment effects for individual shoppers.

The experimental unit is a store-day. For store-days in the October–December 2026 schedule, using the checkout promotion causes an increase in mean daily sales. The conclusion is at the store-day level, not an independently randomized shopper-level effect.

- Distinguish making groups similar from assigning exposure by chance. - Identify variables that may not have been included in the match. - Phrase the conclusion as an association when self-selection remains.

1. Matching can make the two groups more similar on the variables used for matching. 2. Meal-delivery use was still self-selected rather than randomly assigned. 3. Unmeasured differences, such as work schedule or interest in cooking, may remain confounded with service use. 4. The study supports an association between service use and less cooking time among the matched people, but not a causal effect.

Matching does not create random assignment. The study supports an association between meal-delivery use and less cooking time among the matched people, but it does not establish that the service caused the reduction.

- Distinguish an overall average effect from effects within subgroups. - Use the randomized design and stated statistical evidence for the overall conclusion only. - Ask whether the study reports separate evidence for the named groups.

1. Random selection, random assignment, and the statistically convincing overall difference support a population-level causal conclusion about the average effect for adult members represented by the sample. 2. An overall mean effect does not show that the effect has the same size or direction in every subgroup. 3. Subgroup conclusions require adequate representation and planned analysis within the relevant age and sex groups. 4. The supported conclusion is an average causal effect for the full member population, not a guaranteed effect for each subgroup or individual.

The study supports that Plan B lowers mean race time on average for the represented adult-member population. It does not prove the same effect for every age group, for both sexes separately, or for every individual without subgroup evidence.

- Compare the proportions of missing outcomes across the two assigned groups. - Ask whether the people who remain are still likely to be comparable. - Distinguish the originally assigned sample from the smaller group included in the final analysis.

1. The original random selection and random assignment would support population-level causal inference if outcome data were retained comparably across groups. 2. Large differential dropout can make the remaining treatment and control groups systematically different. 3. A comparison of completers may therefore reflect both the treatment and differences in who remained. 4. The clean causal conclusion for all sampled employees, and its generalization to all employees, is weakened unless the missing outcomes are addressed appropriately.

Differential attrition can destroy the comparability created by random assignment among the analyzed completers. A complete-case comparison may not support a causal conclusion for all sampled employees or a generalization to all employees without addressing the missing outcomes.