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A school district introduces an optional after-school tutoring program. At the end of the semester, students who attended tutoring at least 10 times had an average final exam score of 86, while those who attended fewer than 10 times averaged 79. No random assignment was used. Which conclusion is most appropriate?
Students who attended tutoring at least 10 times scored higher on average, but the difference could be influenced by factors like motivation or prior performance.
The data show that tutoring has no effect, because students who need tutoring are weaker and would otherwise score even lower.
Attending tutoring at least 10 times caused the 7-point increase, so requiring tutoring would raise every student’s final exam score by 7 points.
Because tutoring was optional, the higher average proves that tutoring is more effective than any in-class instruction for all students.
Explanation
This question determines the most appropriate conclusion from optional tutoring data without random assignment. The data show attendees (10+ times) averaged 86, higher than 79 for others, indicating an association. Choice B is supported as it notes the difference but cautions about confounders like motivation. Choice A oversteps by claiming causation and universal effects. The observational design limits isolating tutoring's impact. When no randomization occurs, highlight potential self-selection biases in interpretations.
A researcher wants to compare two methods of memorizing vocabulary. She recruits 24 students from one language club and randomly assigns 12 to Method 1 and 12 to Method 2. After one week, Method 1 students average 18.5 words correct out of 25, and Method 2 students average 16.0. Which statement is best supported?
Method 1 is proven to be better for all students everywhere, because random assignment guarantees results apply to any population.
The difference must be due to Method 2 being used incorrectly, because random assignment eliminates all possibility of random variation.
Method 1 caused higher average scores than Method 2 for these club students under the study conditions, but generalizing beyond similar students should be done cautiously.
No conclusion can be made because random assignment prevents comparing group averages, so the means are not meaningful.
Explanation
The question assesses the best-supported statement from a randomized vocabulary method experiment in club students. The data show Method 1 averaged 18.5 words correct vs. 16.0 for Method 2, supporting superiority in this context. Choice A is appropriate as it claims causation but cautions generalization beyond similar students. Choice B overgeneralizes to all students universally. Randomization supports comparison. For small, specific samples, emphasize limited scope despite strong design.
A teacher reports that students who sit in the front two rows have an average course grade of 88, while students who sit elsewhere average 81. Seating is chosen by students on the first day. The teacher claims, “Sitting in front improves grades.” Which statement is most appropriate?
The data show there is no relationship between seating and grades, because students can choose seats and therefore the groups are identical.
The data prove sitting in front causes grades to rise by 7 points, since the averages differ and the course is the same for everyone.
The data prove that higher grades cause students to sit in the front, so moving students forward would lower their grades.
The higher average among front-row students could reflect differences in motivation or prior achievement, so the data do not justify a causal claim about seating.
Explanation
The question evaluates a teacher's claim from seating and grade data, where students chose seats. The data show front-row averages of 88 vs. 81 elsewhere, indicating an association. Choice A is correct as it suggests confounders like motivation, limiting causation. Choice B claims proven causation, overstepping self-selection. Direction is unclear. Without randomization, note that associations may reflect underlying differences.
A sports scientist randomly assigns 60 runners to either a new training plan or their usual plan for 8 weeks. Average 5K time (minutes) decreases by 1.4 minutes in the new-plan group and by 0.6 minutes in the usual-plan group. Which claim is best supported?
The new training plan caused a greater average improvement in 5K time than the usual plan for these runners, given random assignment and the observed mean changes.
The new training plan will improve every runner’s 5K time by exactly 1.4 minutes, so it is guaranteed to work equally well for all athletes.
The data show only that faster runners were placed into the new-plan group, so the plan cannot have caused any improvement.
Because both groups improved, the new plan had no effect compared with usual training, and the 0.8-minute difference must be due to chance only.
Explanation
This question identifies the best-supported claim from a randomized training plan experiment on runners. The data show greater average improvement (1.4 vs. 0.6 minutes) for the new plan, supporting causation in this group. Choice A is justified as it attributes the difference to the plan while noting scope. Choice B overgeneralizes to all runners with exact effects. Both improved, but the difference matters. In experiments, highlight causal support but cautious generalization.
A student wants to know whether students at her school prefer paper textbooks or digital textbooks. She asks the first 80 students entering the library on Monday morning and finds 55 prefer digital. Which statement best describes what can be concluded?
The sample suggests many surveyed library visitors prefer digital textbooks, but the sampling method may not represent all students at the school.
Because 55 out of 80 is a majority, most students at the school prefer digital textbooks, regardless of who was sampled.
The result shows that exactly $\frac{55}{80}$ of students at the school prefer digital textbooks, because the sample size is sufficiently large.
The result proves that switching to digital textbooks will increase student learning, since preference implies effectiveness.
Explanation
The question describes what can be concluded from a convenience sample of 80 library entrants, where 55 preferred digital textbooks. The data suggest a preference in this group, but the method may bias toward library users. Choice A is appropriate as it notes the suggestion but questions representativeness. Choice B overgeneralizes to all students despite sampling issues. Preference does not imply effectiveness. For non-random samples, limit claims to the sampled group and note biases.
An environmental group measures air quality index (AQI) and the number of asthma-related ER visits in one city for 24 months. Months with higher AQI tend to have more ER visits. The group claims, “Reducing AQI will reduce asthma ER visits.” Which statement is most justified?
The data prove that high AQI is the only cause of asthma ER visits, since the trend appears across 24 months.
The data show an association over time in that city, but other seasonal factors could affect both AQI and ER visits, so causation is not certain.
The data prove that asthma ER visits cause AQI to rise, because ER visits increase in months with higher AQI.
The claim is fully justified for all cities, because a two-year study in one city automatically generalizes to every location.
Explanation
The question evaluates an environmental group's claim from 24 months of AQI and ER visit data in one city. The data show months with higher AQI had more visits, indicating a temporal association. Choice A is justified as it acknowledges the link but notes potential seasonal confounders limiting causation. Choice B claims sole causation, overstepping the evidence. Generalization to all cities is unsupported. In time-series data, distinguish correlations from proven causal mechanisms.
A researcher studies whether a new helmet reduces concussion risk in youth soccer. Two teams choose to adopt the helmet, and two similar teams do not. Over a season, concussion rates are 1.2 per 100 player-games for helmet teams and 2.0 per 100 player-games for non-helmet teams. Which conclusion is most appropriate?
Helmet teams had a lower observed concussion rate, but without random assignment other differences between teams could explain the association.
The helmet definitely caused the lower concussion rate, because the rates differ and the teams are described as similar.
The results prove that exactly 0.8 concussions per 100 player-games will be prevented for every team that adopts the helmet.
The helmet increases concussion risk, because teams choosing helmets may be more aggressive and still had some concussions.
Explanation
This question assesses the most appropriate conclusion from observational helmet data in soccer teams. The data show lower concussion rates (1.2 vs. 2.0 per 100 games) for helmet teams, indicating an association. Choice B is supported as it notes the association but highlights lack of randomization allowing team differences. Choice A claims definite causation, overstepping the design. Exact prevention rates are unproven. Without experiments, emphasize that observations suggest but do not confirm causes.
A manager tests whether playing music increases productivity by measuring the number of items packed per hour in a warehouse. For one week, music is played each day; for the next week, no music is played. The average packed per hour is 102 with music and 95 without music. Which issue most limits concluding that music caused the increase?
Since productivity is measured per hour, the results cannot be compared across weeks, so no association can be described.
The conclusion is fully justified because using the same warehouse controls all variables, so causation is guaranteed.
The weeks were not randomized or alternated, so other changes between weeks (orders, staffing, learning effects) could explain the difference.
Because the averages differ by 7, the increase must be caused by music rather than any other factor.
Explanation
The question identifies the key limitation in concluding music caused a productivity increase in a warehouse test. The data show averages of 102 with music and 95 without, but sequential weeks may introduce confounders. Choice A is correct as non-randomized timing allows other factors like learning effects. Choice B assumes causation without addressing design flaws. Same location does not control temporal changes. In time-based studies, consider sequencing biases that confound results.
A nutritionist measures sugar intake and body mass index (BMI) for 150 clients at a weight-loss clinic. The nutritionist finds a positive association and states, “Eating sugar increases BMI.” Which statement is best supported by the information given?
The data imply there is no relationship between sugar intake and BMI, because clinic clients are not representative of the general population.
The data prove sugar intake causes higher BMI for all people, since 150 is a large enough sample to establish causation.
The data show a positive association among clinic clients, but a causal claim is not justified because this is an observational sample with potential confounders.
The data prove that higher BMI causes people to eat more sugar, so reducing BMI would automatically reduce sugar intake.
Explanation
The question evaluates a nutritionist's claim from positive association data on sugar intake and BMI in 150 clinic clients. The data indicate higher sugar aligns with higher BMI in this group, but causation is unproven. Choice A is justified as it describes the association while noting observational limits and confounders. Choice B claims universal causation, overstepping the sample's scope. Direction of causation is ambiguous. For correlations in biased samples, stress distinctions between association and proof of cause.
A school surveyed 120 students who visited the cafeteria at least once last week about whether they ate breakfast daily and their math test score (out of 100) from the most recent exam. The survey was voluntary and conducted during lunch. Results are summarized below.
Table: Average math score by breakfast habit
- Eats breakfast daily: $n=70$, mean $=84$
- Does not eat breakfast daily: $n=50$, mean $=78$
Which conclusion is most appropriate based on the study design and data?
Because the sample size is over 100, the difference in averages must be due to breakfast rather than other factors.
Eating breakfast daily causes students’ math scores to increase by about 6 points, so requiring breakfast would raise scores for all students.
Among surveyed cafeteria visitors, students reporting daily breakfast had a higher average math score than those not reporting daily breakfast.
All students at the school who eat breakfast daily score higher in math than all students who do not eat breakfast daily.
Explanation
This question asks which conclusion is most appropriate given the study's design and data from a voluntary survey of cafeteria visitors. The data show that among the 120 surveyed students, those reporting daily breakfast (n=70) had an average math score of 84, while those not (n=50) averaged 78, indicating an association in this sample. Choice B is supported because it accurately describes the observed difference without claiming causation or generalizing beyond the surveyed group. In contrast, choice A oversteps by assuming causation and suggesting a policy like requiring breakfast, which the observational data cannot prove. Similarly, choices C and D overgeneralize or misattribute the cause, ignoring potential biases like self-selection in the voluntary survey. When evaluating claims from surveys, always distinguish between observed associations in a sample and unproven causal effects or population-wide truths.