- A company randomly assigned 70 employees to two email-subject-line styles for an internal newsletter: Treatment A (question-style subject lines) and Treatment B (statement-style subject lines), 35 employees per group. The outcome was the time (in seconds) to open the email after it was sent.
Results: Treatment A mean open time = 52 s, Treatment B mean open time = 55 s. Observed difference in means (A − B) = −3 s.
To test whether Treatment A leads to faster opening times, the company conducted a randomization test by shuffling the treatment labels 2,000 times and recomputing (A − B). In the simulation, 1,540 of the 2,000 shuffled differences were at most −3 s (i.e., as negative as observed).
Which conclusion is most reasonable about the treatment effect?
Answer: There is not enough evidence that Treatment A makes open times faster, because differences at least as negative as −3 seconds happened often (1,540/2,000) under no effect.
Randomized experiments and simulations help compare treatments by assessing if outcomes differ more than expected by chance. Random assignment ensures fair groups, supporting that treatments cause observed effects. The observed difference is -3 seconds in mean open times, with A faster than B. The 'no effect' simulation shuffles labels repeatedly, producing a distribution of possible differences from randomness alone. 1,540 out of 2,000 simulations showed differences as negative as or more than -3, meaning it's common under no effect. Don't confuse random assignment with random sampling; it aids causal inference, and rarity suggests evidence, not proof. For other cases, look at how often simulations yield differences as extreme as observed. - In a randomized experiment, 90 participants were randomly assigned (45 each) to two different instruction videos: Treatment A (interactive) and Treatment B (standard). After viewing, each participant answered 10 comprehension questions. Treatment A had a mean score of 7.1 and Treatment B had a mean score of 6.2, so the observed difference in means was 7.1−6.2=0.9 (A − B). A randomization test shuffled the A/B labels 1000 times under "no treatment effect." In the 1000 shuffles, 8 produced a difference in means of at least +0.9 (A − B).
Based on the randomization test, is the observed difference surprising under no effect?
Answer: Yes. Since only 8 out of 1000 shuffles were at least +0.9, the observed difference would be unusual under no effect; there is evidence the interactive video increases the mean score.
This question deals with comparing instructional videos in a randomized experiment and simulation to check if an interactive one boosts comprehension scores. Random assignment equalizes groups on other factors, permitting cause-and-effect conclusions from treatment differences. The observed difference is in mean scores, 0.9 points higher for Treatment A than B. The 'no effect' simulation shuffles labels on fixed scores many times, forming a distribution of chance-based differences assuming no treatment impact. With just 8 out of 1000 shuffles at least +0.9, the observed difference is surprising under no effect, indicating evidence for Treatment A's benefit. A key misconception is confusing random assignment with random sampling; the first aids causality in the experiment, and rarity supports but doesn't prove the effect universally. Apply this by seeing the proportion of extreme simulated differences to determine rarity.
- A coach randomly assigned 72 athletes to two warm-up routines before a sprint test: Treatment A (dynamic warm-up) and Treatment B (light jogging). There were 36 athletes per group. The outcome was sprint time in seconds (lower is better).
Treatment A mean time = 12.84 s, Treatment B mean time = 12.90 s. The observed difference (A − B) is −0.06 s.
A randomization test was performed by shuffling the warm-up labels 1,000 times under "no treatment effect." In the simulations, 463 of the 1,000 shuffled differences were at most −0.06 (i.e., at least as negative as the observed difference, in the direction favoring A).
Which conclusion is most reasonable about whether Treatment A leads to faster times than Treatment B (one-sided, more negative is more extreme)?
Answer: Because 463 out of 1,000 simulated differences were at most −0.06, the observed difference is not unusual under no effect, so there is not strong evidence that Treatment A makes athletes faster.
In comparing treatments using a randomized experiment and simulation, we explore if a dynamic warm-up (Treatment A) leads to faster sprint times than light jogging (Treatment B), where lower times are better. Random assignment balances potential confounders across groups, allowing us to conclude that observed differences are likely caused by the treatments. The observed difference is A's mean time minus B's, which is -0.06 seconds here, with more negative values favoring A. The 'no effect' simulation reassigns treatments randomly 1,000 times, creating a distribution of differences possible if the warm-ups had no true effect. Here, 463 out of 1,000 simulated differences were at most -0.06, showing this isn't rare under no effect, so there's not strong evidence for A's superiority. A misconception is equating random assignment with random sampling—assignment supports causality here, but sampling would generalize broadly; rarity provides evidence, but common results don't disprove effects entirely. To transfer this, examine how often simulations produce differences as extreme (in the relevant direction) as the observed one. - A teacher randomly assigned 90 students to two different note-taking methods during a lecture: Treatment A (guided notes) and Treatment B (blank paper). There were 45 students in each group. After the lecture, students took a 20-question quiz.
Treatment A mean = 16.8, Treatment B mean = 15.1, so the observed difference (A − B) is 1.7 questions.
A randomization test shuffled the 90 quiz scores between groups 1,000 times under "no treatment effect." In the simulations, 3 of the 1,000 shuffled differences were at least as large as 1.7.
Which conclusion is most reasonable about the treatment effect (one-sided, A − B)?
Answer: Because only 3 out of 1,000 simulated differences were at least as large as 1.7, the observed difference is unlikely under no effect, so the data provide evidence that guided notes increase the mean quiz score.
In comparing treatments using a randomized experiment and simulation, we determine if guided notes (Treatment A) yield higher quiz scores than blank paper (Treatment B) during lectures. Random assignment helps eliminate other influences, making it reasonable to attribute score differences to the note-taking methods. The observed difference is A's mean score minus B's, which is 1.7 questions here. The 'no effect' simulation mixes up the scores between groups 1,000 times, forming a distribution of differences that could arise by chance if methods don't differ. Only 3 out of 1,000 simulated differences were at least as large as 1.7, marking the observed as rare under no effect and offering evidence for A's benefit. Don't confuse random assignment with random sampling; assignment enables causal claims in this study, while sampling generalizes to populations, and rarity suggests but doesn't prove causation. For other experiments, tally the simulated differences that match or exceed the observed extremity to gauge surprise. - A school randomly assigned 60 students to use either Treatment A (a spaced-practice study plan) or Treatment B (a cramming study plan) for one week before a quiz. Treatment A had n=30 with a mean quiz score of 82 points; Treatment B had n=30 with a mean quiz score of 74 points. The observed difference in means was 82−74=8 points (A − B). To test whether this difference could be due to chance under "no treatment effect," the teacher performed a randomization test by shuffling the treatment labels 1,000 times and recalculating (A − B) each time. In the simulation, 9 of the 1,000 shuffled differences were at least as large as 8 points.
Which conclusion is most reasonable about the treatment effect?
Answer: Because only 9 of 1,000 shuffled differences were 8, the observed difference would be rare if there were no treatment effect, so the results provide evidence that Treatment A increases mean quiz score compared with Treatment B.
In comparing treatments using a randomized experiment and simulation, we investigate if one treatment performs better than another by assigning participants randomly to groups. Random assignment helps balance out other factors, allowing us to attribute differences in outcomes to the treatments themselves rather than chance or biases, supporting cause-and-effect conclusions. The observed difference here is the difference in mean quiz scores between Treatment A and B, which was 8 points higher for A. The 'no effect' simulation shuffles the treatment labels many times, assuming treatments make no difference, and creates a distribution of possible differences that could occur just by chance. By comparing the observed 8-point difference to this simulated distribution, we see that only 9 out of 1,000 shuffled differences were as large as or larger than 8, indicating the observed result is rare under no effect. A common misconception is that random assignment is the same as random sampling from a population, but it actually helps with causal inference within the study group, and rarity provides evidence but not absolute proof of an effect. To apply this strategy elsewhere, count how often simulated differences are at least as extreme as the observed one to assess if the result is surprising. - A company randomly assigned 50 customer-service trainees to one of two training modules: Treatment A (interactive practice) and Treatment B (video-only). There were 25 trainees per group. After training, each trainee completed a timed task; the outcome was the number of correct entries in 10 minutes.
Treatment A mean = 27.4 correct entries, Treatment B mean = 24.9 correct entries, so the observed difference (A − B) is 2.5.
To check if this could happen by chance under "no treatment effect," the company shuffled the treatment labels 5,000 times and computed (A − B) each time. In the shuffled results, 7 of the 5,000 differences were at least as large as 2.5.
Do the results provide evidence that Treatment A is more effective than Treatment B (one-sided, A − B)?
Answer: Yes. Because only 7 out of 5,000 simulated differences were at least as large as 2.5, the observed difference would be rare under no effect, so the data suggest Treatment A increases the mean number of correct entries.
In comparing treatments using a randomized experiment and simulation, we assess if interactive practice (Treatment A) results in more correct entries than video-only training (Treatment B) for customer-service trainees. Random assignment ensures groups are similar on average, so differences in outcomes can be attributed to the treatments, supporting cause-and-effect conclusions. The observed difference is the mean correct entries for A minus B, equaling 2.5 here. The 'no effect' simulation shuffles treatment assignments 5,000 times, generating a distribution of differences that might occur by random chance alone if treatments don't matter. We find only 7 out of 5,000 simulated differences were at least as large as 2.5, indicating the observed result is rare under no effect and suggesting evidence for A's effectiveness. Note that random assignment differs from random sampling; it allows causal inference in this group but doesn't extend to all trainees, and rarity offers evidence, not absolute proof. In similar scenarios, check the frequency of simulated differences equaling or exceeding the observed to evaluate rarity. - A coach randomly assigned 40 basketball players to two free-throw practice routines for one week: Treatment A (short daily sessions) and Treatment B (one long weekly session). Twenty players were randomly assigned to each routine. At the end of the week, the mean number of made free throws out of 20 was 15.1 for Treatment A and 13.4 for Treatment B, so the observed difference in means was 15.1−13.4=1.7 made shots (A − B). The coach ran a randomization test by shuffling the routine labels 1000 times and recalculating (A − B). In the 1000 shuffles, 6 simulated differences were 1.7.
Which conclusion is most reasonable about the treatment effect?
Answer: There is evidence that Treatment A increases the mean number of made free throws, because a difference at least as large as 1.7 occurred very rarely in the shuffles.
This problem examines basketball practice routines through a randomized experiment. Players were randomly assigned to short daily sessions or one long weekly session, enabling causal conclusions about free-throw performance. The observed difference in means was 1.7 made shots favoring daily practice. To assess if this could happen by chance, 1000 random reassignments were simulated, with only 6 producing differences at least as large as 1.7. Since 6 out of 1000 is very rare (0.6%), the observed difference is surprising under "no effect," providing strong evidence that daily practice increases free-throw success. A key point: random assignment (which did occur) allows causal conclusions, while the misconception in option D incorrectly claims players chose their routine.
- A community center randomly assigned 72 participants to two ways of learning a new card game: Treatment A (in-person demonstration) and Treatment B (instruction booklet). There were 36 participants in each group. The outcome was whether the participant could correctly play a full round without help (success/failure). In Treatment A, 27 of 36 succeeded (p^A=0.75). In Treatment B, 18 of 36 succeeded (p^B=0.50). The observed difference in proportions was 0.75−0.50=0.25 (A − B). A randomization test shuffled the success/failure outcomes across the two groups 4000 times under "no treatment effect." In the simulations, 40 of the 4000 shuffled differences were at least as large as 0.25 (A − B).
Which conclusion is most reasonable about the treatment effect?
Answer: There is evidence that the in-person demonstration increases the probability of success, because only 40 out of 4000 shuffled differences were at least 0.25, which is rare under no effect.
This problem involves comparing treatments using proportions in a randomized experiment. Random assignment of participants allows causal conclusions about which learning method increases success probability. The observed difference of 0.25 (75% minus 50%) shows the in-person demonstration had a 25 percentage point higher success rate. The randomization test checks if this could occur by chance under no effect by shuffling outcomes 4000 times. Finding only 40 out of 4000 shuffled differences at least as large as 0.25 means the observed difference would be rare (1%) under no effect. This rarity provides evidence that the demonstration increases success probability, though it doesn't guarantee a 25-point increase for every individual. Randomization tests work perfectly well with proportions, not just means. When checking if A is better than B, we look for simulated differences at least as large as observed in the same direction (positive when A−B is calculated).
- In a randomized experiment, 60 students were randomly assigned by shuffling identical cards (30 labeled A and 30 labeled B) to either Treatment A (a new study app) or Treatment B (a standard app). After one week, each student took the same 20-question quiz. The mean score for Treatment A was 16.4 and for Treatment B was 14.9, so the observed difference in means was 16.4−14.9=1.5 points (A − B). To test whether this difference could be due to chance, the researcher performed a randomization test by keeping all 60 quiz scores fixed and repeatedly shuffling the A/B labels 2000 times under "no treatment effect." In the 2000 shuffles, 18 produced a difference in means of at least +1.5 points (A − B).
Which conclusion is most reasonable about the treatment effect (A − B)?
Answer: Because only 18 out of 2000 shuffles produced a difference at least as large as +1.5, the observed difference would be unusual under no effect, so there is evidence Treatment A increases the mean quiz score compared with Treatment B.
This question explores comparing treatments using a randomized experiment and simulation to assess if a new study app improves quiz scores. Random assignment of students to treatments helps ensure that any observed differences are likely due to the treatments themselves, allowing for cause-and-effect conclusions by balancing out other factors across groups. The observed difference is the difference in mean quiz scores, here 1.5 points higher for Treatment A than B. The 'no effect' simulation shuffles the treatment labels many times while keeping scores fixed, creating a distribution of possible differences that could occur just by chance if the treatments had no real impact. By comparing the observed +1.5 to this distribution, we see that only 18 out of 2000 shuffles were at least as extreme, indicating the result is rare under no effect and provides evidence of a treatment effect. A common misconception is that random assignment is the same as random sampling from a population, but it actually supports causal claims within the study group, and rarity offers evidence but not absolute proof. To apply this strategy elsewhere, check how often simulated differences are as extreme or more than the observed one to gauge surprise under no effect.
- A company tested two website designs to see which leads to more newsletter sign-ups. They randomly assigned 80 visitors to Treatment A (new design) and 80 visitors to Treatment B (old design). In Treatment A, 44 of 80 visitors signed up (p^A=0.55). In Treatment B, 28 of 80 visitors signed up (p^B=0.35). The observed difference in proportions was 0.55−0.35=0.20 (A − B). Under the assumption of no treatment effect, the analyst shuffled the "A/B" labels 5000 times and computed the difference in sign-up rates each time. In the 5000 shuffles, 6 differences were at least as large as 0.20 (A − B).
Based on the randomization test, is the observed difference surprising under no effect?
Answer: Yes. Only 6 out of 5000 shuffled differences were at least 0.20, so the observed difference would be rare under no effect, suggesting the new design increases sign-ups.
This problem involves comparing website designs using a randomized experiment with proportions. Random assignment of visitors to designs A and B allows causal conclusions about which design increases sign-ups. The observed difference of 0.20 (55% minus 35%) shows Treatment A had a 20 percentage point higher sign-up rate. The randomization test checks if this difference could occur by chance under no treatment effect by shuffling the A/B labels 5000 times. Finding only 6 out of 5000 shuffled differences at least as large as 0.20 means the observed difference would be very rare (about 0.1%) if the designs had no effect. This rarity provides strong evidence that the new design increases sign-ups. The sample size of 160 doesn't automatically make differences significant—we need to check rarity through simulation. A common misconception is thinking 6 occurrences means the difference is common, when actually 6/5000 indicates extreme rarity.
- In a randomized experiment, 72 volunteers were randomly assigned to two different puzzle-solving strategies: Treatment A (work backward) and Treatment B (trial-and-error), 36 per group. The outcome was the number of hints used (lower is better). Treatment A had a mean of 2.1 hints and Treatment B had a mean of 2.9 hints, so the observed difference in means was 2.1−2.9=−0.8 (A − B). A randomization test shuffled the treatment labels 2500 times under "no treatment effect." In the 2500 shuffles, 5 produced a difference in means of at most −0.8 (A − B).
Which conclusion is most reasonable about the treatment effect (A − B), where more negative values indicate fewer hints for Treatment A?
Answer: Because only 5 out of 2500 shuffles were at most −0.8, the observed difference would be unusual under no effect; there is evidence Treatment A reduces the mean number of hints used compared with Treatment B.
This question examines comparing puzzle strategies in a randomized experiment and simulation to check if working backward reduces hints needed. Random assignment minimizes differences between groups, supporting cause-and-effect conclusions from treatments. The observed difference is in mean hints, -0.8 for Treatment A minus B, showing fewer for A. The 'no effect' simulation shuffles labels on fixed hint counts many times, building a distribution of chance differences assuming no treatment effect. Only 5 out of 2500 shuffles were at most -0.8, indicating the observed is unusual and evidencing Treatment A's benefit. A misconception is equating random assignment to random sampling; it facilitates causality in this context, with rarity providing evidence, not certainty. For transfer, focus on how often simulations are as extreme or more than observed.
- A library tested two reminder methods for returning books on time. They randomly assigned 50 patrons to Treatment A (text reminder) and 50 patrons to Treatment B (email reminder). The outcome was whether the book was returned on time. Treatment A had 41 on-time returns (0.82) and Treatment B had 33 on-time returns (0.66). The observed difference in proportions was 0.82−0.66=0.16 (A − B). A randomization test was performed by shuffling the on-time/late outcomes among all 100 patrons 4000 times under "no treatment effect." In the 4000 shuffles, 12 produced a difference in proportions of at least +0.16 (A − B).
Which conclusion is most reasonable about the treatment effect (A − B)?
Answer: Because only 12 out of 4000 shuffles were at least +0.16, the observed difference would be unusual under no effect; there is evidence text reminders increase the on-time return proportion compared with email reminders.
This question covers comparing reminder methods in a randomized experiment and simulation to see if texts improve on-time book returns over emails. Random assignment makes groups similar, allowing attribution of differences to treatments for cause-and-effect inferences. The observed difference is in return proportions, 0.16 higher for Treatment A than B. The 'no effect' simulation shuffles outcomes across patrons multiple times, generating a distribution of possible differences by chance alone under no effect. Only 12 out of 4000 shuffles were at least +0.16, meaning the observed is unusual and provides evidence for Treatment A's effectiveness. Remember, random assignment supports causality in the study unlike random sampling for generalization, and rarity offers evidence without absolute proof. To use this approach, evaluate the count of simulations as or more extreme than observed.
- A music instructor randomly assigned 48 beginners to two practice strategies for learning a short piece: Treatment A (practice with a metronome) and Treatment B (practice without a metronome). Twenty-four students were randomly assigned to each strategy. After one week, the mean performance rating (0–10 scale) was 7.3 for Treatment A and 6.5 for Treatment B, so the observed difference in means was 7.3−6.5=0.8 (A − B). The instructor performed a randomization test by shuffling the strategy labels 2000 times and recalculating (A − B). In the 2000 shuffles, 14 simulated differences were 0.8.
Which conclusion is most reasonable about the treatment effect?
Answer: There is evidence that practicing with a metronome increases the mean rating, because differences at least as large as 0.8 were rare in the shuffles.
This question examines music practice strategies using a randomized experiment. Students were randomly assigned to practice with or without a metronome, allowing causal conclusions about performance ratings. The observed difference in means was 0.8 points favoring metronome practice. The randomization test simulated 2000 random reassignments and found only 14 produced differences at least as large as 0.8. Since 14 out of 2000 is rare (0.7%), the observed difference is surprising under "no effect," providing evidence that metronome practice increases performance ratings. A common misconception is thinking rare results in simulations mean the difference is "typical" or due to confounding—actually, rarity suggests the treatment effect is real, not due to chance or hidden variables.
- A school randomly assigned 60 students to try two different study plans for a vocabulary quiz: Treatment A (spaced practice) and Treatment B (single long review). Each group had 30 students. The quiz was scored out of 50 points. Treatment A had a mean score of 41.2 and Treatment B had a mean score of 38.6, for an observed difference (A − B) of 2.6 points. To test whether this difference could be due to chance under "no treatment effect," the teacher performed a randomization test by shuffling the treatment labels 1,000 times and recalculating (A − B) each time. In the simulations, 12 of the 1,000 shuffled differences were at least as large as 2.6.
Which conclusion is most reasonable about the treatment effect (using the one-sided direction A − B)?
Answer: Because only 12 out of 1,000 simulated differences were at least as large as 2.6, the result would be rare under no effect, so the data provide evidence that Treatment A leads to higher mean scores than Treatment B.
In comparing treatments using a randomized experiment and simulation, we investigate if spaced practice (Treatment A) leads to higher vocabulary quiz scores than a single long review (Treatment B). Random assignment of students to treatments helps ensure that any observed difference in scores is likely due to the treatments themselves, not other factors, allowing us to draw cause-and-effect conclusions. The observed difference is the mean score for A minus the mean for B, which here is 2.6 points. The 'no effect' simulation shuffles the treatment labels many times, assuming the treatments make no difference, and creates a distribution of possible differences that could occur just by chance. We compare the observed difference of 2.6 to this simulated distribution and find that only 12 out of 1,000 simulated differences were at least as large, making the observed result rare under no effect and providing evidence for a treatment effect. A common misconception is that random assignment is the same as random sampling, but random assignment balances groups in the experiment while random sampling would generalize to a larger population; also, rarity provides evidence but doesn't prove the effect beyond doubt. To apply this strategy elsewhere, count how often simulated differences are as extreme as or more extreme than the observed one to assess if the result is surprising. - A class tested two note-taking methods. The teacher randomly assigned 28 students to Treatment A (guided notes) and 28 students to Treatment B (blank notes). After a lesson, students took the same 12-question quiz. Treatment A's mean score was 9.8 and Treatment B's mean score was 9.1, so the observed difference in means was 9.8−9.1=0.7 (A − B). A randomization test shuffled the A/B labels 1500 times under "no treatment effect." In the 1500 shuffles, 735 produced a difference in means of at least +0.7 (A − B).
Do the results provide evidence that Treatment A is more effective than Treatment B?
Answer: No. Since 735 out of 1500 shuffles were at least +0.7, a difference like +0.7 is fairly common under no effect; there is not strong evidence that guided notes increase the mean quiz score.
This question addresses comparing note-taking methods in a randomized experiment with simulation to determine if guided notes enhance quiz scores. Through random assignment, groups are balanced, enabling cause-and-effect conclusions about treatments. The observed difference is in mean scores, 0.7 points higher for Treatment A than B. The 'no effect' simulation shuffles labels on fixed scores repeatedly, producing a distribution of differences expected by chance if treatments had no influence. With 735 out of 1500 shuffles at least +0.7, the difference is fairly common under no effect, indicating not strong evidence for Treatment A. It's important not to mix random assignment with random sampling; the former allows causal claims here, and rarity suggests evidence but isn't proof. In similar analyses, check the frequency of simulated differences as extreme as observed.
- A principal randomly assigned 40 classrooms to two ways of organizing a 10-minute daily reading time for one month: Treatment A (silent independent reading) and Treatment B (teacher read-aloud). Twenty classrooms were randomly assigned to each method. The outcome was the mean increase in a reading fluency score (points) from pretest to posttest. Treatment A had a mean increase of 12.3 points and Treatment B had a mean increase of 10.9 points, so the observed difference in means was 12.3−10.9=1.4 points (A − B). Under "no treatment effect," the principal shuffled the treatment labels across classrooms 2000 times and recalculated the difference each time. In the simulations, 52 of the 2000 shuffled differences were at least as large as 1.4 (A − B).
Which conclusion is most reasonable about the treatment effect?
Answer: Because only 52 out of 2000 shuffled differences were at least 1.4, the observed difference is relatively rare under no effect, providing evidence that Treatment A tends to produce a larger mean increase than Treatment B.
This question involves interpreting borderline randomization test results. Random assignment of classrooms allows us to attribute differences to the reading method rather than classroom characteristics. The observed difference of 1.4 points (12.3 minus 10.9) shows independent reading had a larger mean increase. The randomization test simulates differences under no effect by shuffling labels 2000 times. Finding 52 out of 2000 shuffled differences at least as large as 1.4 means the observed difference would occur about 2.6% of the time by chance. This relative rarity (less than 5%) provides evidence that independent reading tends to produce larger gains, though the evidence isn't extremely strong. Random assignment actually does balance confounders on average, contrary to option C. The result doesn't guarantee every classroom gains exactly 1.4 points—it suggests a tendency for Treatment A to be more effective on average.
- In a randomized experiment, 48 students were randomly assigned to two different methods for learning keyboard shortcuts: Treatment A (game-based practice) and Treatment B (worksheet practice). There were 24 students per group. After one week, the outcome was the number of shortcuts correctly recalled (out of 30).
Treatment A mean = 21.5, Treatment B mean = 18.9, so the observed difference (A − B) is 2.6.
A randomization test was conducted by shuffling the 48 recall scores between groups 2,000 times under "no treatment effect." In the simulations, 28 of the 2,000 shuffled differences were at least as large as 2.6.
Which conclusion is most reasonable about the treatment effect (one-sided, A − B)?
Answer: Because 28 out of 2,000 simulated differences were at least as large as 2.6, the result would be rare under no effect, so the data suggest Treatment A increases the mean number of shortcuts recalled.
In comparing treatments using a randomized experiment and simulation, we evaluate if game-based practice (Treatment A) boosts keyboard shortcut recall more than worksheet practice (Treatment B). Random assignment creates comparable groups, so differences can be linked causally to the methods. The observed difference is A's mean recall minus B's, equaling 2.6 here. The 'no effect' simulation reassigns scores 2,000 times, building a distribution of differences from random variation alone. Only 28 out of 2,000 were at least as large as 2.6, showing rarity under no effect and evidence for A's advantage. Remember, random assignment differs from random sampling; it allows causal conclusions here but not population-wide, and rarity provides supportive evidence, not proof. In other cases, look at the count of simulated extremes to judge if the observed difference is rare. - A teacher randomly assigned 60 students to try two different study plans for a history quiz: Treatment A (spaced practice) and Treatment B (rereading). Thirty students were randomly assigned to each plan. The mean quiz score was 82.4 for Treatment A and 76.1 for Treatment B, so the observed difference in means was 82.4−76.1=6.3 points (A − B). To test whether this difference could be due to chance under "no treatment effect," the teacher performed a randomization test by shuffling the study-plan labels 2000 times and recalculating (A − B) each time. In the 2000 shuffles, 18 simulated differences were at least as large as 6.3.
Which conclusion is most reasonable about the treatment effect?
Answer: There is evidence that Treatment A leads to a higher mean quiz score than Treatment B, because a difference at least as large as 6.3 points was rare in the shuffles.
This question asks us to compare treatments using a randomized experiment and simulation. Because students were randomly assigned to Treatment A (spaced practice) or Treatment B (rereading), we can draw cause-and-effect conclusions about which treatment leads to better quiz scores. The observed difference in means was 6.3 points favoring Treatment A. To test if this could happen by chance under "no treatment effect," the teacher simulated 2000 random reassignments and found only 18 produced differences at least as large as 6.3. Since 18 out of 2000 is rare (less than 1%), the observed difference is surprising under no effect, providing evidence that Treatment A causes higher scores. Note that random assignment (not random sampling) allows causal conclusions, and evidence supports but doesn't prove the effect.
- A researcher randomly assigned 100 houseplants to two fertilizers: Treatment A and Treatment B (50 plants each). After 4 weeks, the outcome was the increase in height (cm). Treatment A had a mean increase of 6.4 cm and Treatment B had a mean increase of 6.0 cm, so the observed difference in means was 6.4−6.0=0.4 cm (A − B). The researcher performed a randomization test by shuffling the fertilizer labels 5000 times and recalculating the difference in means each time. In the simulations, 2400 of the 5000 shuffled differences were at least as large as 0.4 cm (A − B).
Do the results provide evidence that Treatment A is more effective than Treatment B?
Answer: No. Because many shuffled differences (2400 out of 5000) were at least as large as 0.4 cm, the observed difference is not rare under no effect, so there is not strong evidence that Treatment A increases mean growth compared with Treatment B.
This question tests recognizing when many simulated differences indicate lack of evidence. Random assignment of plants to fertilizers allows us to attribute differences to the treatment rather than other factors. The observed difference of 0.4 cm (6.4 minus 6.0) shows Treatment A had slightly more growth on average. The randomization test simulates differences under no treatment effect by shuffling labels 5000 times. Finding 2400 out of 5000 shuffled differences at least as large as 0.4 means the observed difference is common—it would occur about 48% of the time by chance alone. This commonness under no effect means we lack strong evidence that Treatment A increases growth. A misconception is thinking 2400 being "large" makes the difference rare; we must consider the proportion (2400/5000 ≈ 48%). Random assignment doesn't make every difference meaningful, and using ratios instead of differences doesn't change the conclusion about treatment effectiveness.
- A music teacher randomly assigned 44 beginners to learn a short piece using either Treatment A (practice with a metronome) or Treatment B (practice without a metronome), 22 students per group. The outcome was performance accuracy (percent of notes played correctly).
Results: Treatment A mean = 91%, Treatment B mean = 84%. Observed difference in means (A − B) = 7 percentage points.
A randomization test was performed by shuffling the treatment labels 1,000 times under "no treatment effect." In the simulation, 12 of the 1,000 shuffled differences were at least as large as 7 percentage points.
Do the results provide evidence that Treatment A is more effective than Treatment B?
Answer: Yes. Since only 12 of 1,000 shuffled differences were 7, the observed difference would be rare under no effect, suggesting Treatment A increases mean accuracy.
In randomized experiments, simulations compare treatments by checking the likelihood of differences under chance. Random assignment levels the playing field, allowing attribution of differences to treatments causally. Here, the observed difference is 7 percentage points higher mean accuracy for Treatment A. The 'no effect' simulation reshuffles labels many times, forming a distribution of chance-based differences. With just 12 out of 1,000 simulations at least as large as 7, the observed is rare if no effect exists. Misconception: random assignment isn't random sampling; it supports causality, and rarity provides evidence without absolute certainty. Transfer by noting the frequency of equally or more extreme simulated differences.
