Inferences & Claims From Statistics
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PSAT Math › Inferences & Claims From Statistics
A scientist runs an experiment on 40 seeds, randomly assigning 20 to a high-light condition and 20 to a low-light condition. After 2 weeks, high-light seeds have a higher germination rate. Which claim is best supported?
Seeds that were already more likely to germinate were assigned to high light, which explains the result.
The results show only correlation because random assignment prevents any conclusions about cause.
High light will increase germination rates for all plant species in all climates because it worked for these seeds.
Higher light exposure likely caused a higher germination rate in this experiment because the light condition was assigned and other conditions were held constant.
Explanation
This question seeks the best-supported claim from higher germination in randomly assigned high-light seeds. The data show a difference under controlled conditions. Choice A is supported as assignment and controls support causation here. Choice B oversteps by generalizing broadly. Choices C and D misinterpret assignment. Randomization aids causal claims in experiments.
A city tested a new bus schedule on 6 routes for one month and compared average rider wait times to the previous month on the same routes. Weather was unusually mild during the test month. Average wait time decreased on 5 of the 6 routes. Which claim is best supported by the data and study design?
The new schedule caused shorter wait times on all city bus routes, so the city should adopt it everywhere immediately.
Because most tested routes improved, the new schedule will reduce wait times every month, regardless of weather conditions.
On the 6 tested routes, average wait time was lower during the test month than the previous month, but other factors like weather could have contributed.
The decrease on 5 routes proves the schedule is effective, since comparing to the previous month eliminates all confounding variables.
Explanation
This question evaluates which claim is best supported by a before-and-after comparison of bus wait times on tested routes, noting external factors like mild weather. The data indicate lower average wait times on 5 of 6 routes during the test month compared to the previous one, but without controls for variables like weather. Choice C is appropriate as it reports the observed decrease while cautioning that other factors could contribute, reflecting the non-experimental design. Choice A overreaches by claiming causation and urging immediate citywide adoption, ignoring potential confounders. Choice D mistakenly assumes the comparison eliminates all variables, which it does not. In such studies, remember that observed changes show what happened but do not prove what caused it without isolating variables.
A school nurse surveyed 120 students from one high school about whether they ate breakfast and whether they felt “alert” in first period that day. Students were not assigned to eat or skip breakfast; they reported their own choices. Results are shown in the table. Based on the study, which conclusion is most appropriate?
(Assume all students answered honestly.)
Eating breakfast causes students to feel alert in first period because the alert rate is higher among breakfast eaters than among non-eaters.
Among the surveyed students, those who ate breakfast were more likely to report feeling alert in first period than those who did not eat breakfast.
Most high school students in the country would feel more alert if they ate breakfast, because the survey included a large sample size of 120 students.
Skipping breakfast increases sleepiness in first period for every student, because the group that skipped breakfast had a lower alert percentage.
Explanation
This question asks which conclusion is appropriate based on an observational survey about breakfast and alertness. The data shows that 72% of breakfast eaters reported feeling alert versus 45% of non-breakfast eaters - a clear difference in the groups surveyed. Choice B correctly states this finding without claiming causation: "those who ate breakfast were more likely to report feeling alert." Choice A incorrectly claims causation ("causes"), which cannot be established from observational data where students self-selected their breakfast behavior. When interpreting survey data, distinguish between observed associations and causal claims.
A scientist studied 24 mice and randomly assigned 12 to receive a new medicine and 12 to receive a placebo. After treatment, the medicine group had a lower average symptom score. Which statement is most justified?
The medicine will reduce symptoms for all animals and humans because it worked for 12 mice.
The placebo group’s higher average proves the placebo caused symptoms to increase.
The medicine group’s lower average proves that mice with fewer symptoms were more likely to be assigned medicine.
The random assignment supports a causal interpretation that the medicine reduced average symptoms for the mice in this experiment.
Explanation
This question asks which statement is most justified from an experiment where mice randomly assigned to medicine had lower average symptom scores than those on placebo. The data show a difference in averages between the two groups of 12 mice each, with random assignment used. Choice A is supported because random assignment supports causal inference within the experiment by minimizing biases, though it is limited to these mice. Choice B oversteps by generalizing to all animals and humans, which the small mouse study does not prove. Choices C and D mistakenly attribute the differences to assignment biases or reverse causation, contradicting the random assignment. Remember to differentiate between experimental evidence for causation in the sample and unproven broader generalizations.
A researcher measured the number of hours 40 college students studied for a midterm and their exam scores. The scatterplot shows a positive trend: students who studied more tended to score higher, though points are spread out. The researcher claims, “Studying more guarantees a higher score for any student.” Which statement best evaluates this claim?
The claim is supported because a positive association means every additional study hour increases every student’s score by the same amount.
The claim is unsupported because the plot shows no relationship at all between study hours and exam score.
The claim is supported because scatterplots can prove causation whenever the trend is upward.
The claim is too strong because the plot suggests association, not a guarantee, and individual scores vary widely among students with similar study hours.
Explanation
This question evaluates a researcher's claim that more studying guarantees higher exam scores based on a scatterplot of 40 students. The plot shows a positive trend with variation, meaning higher study hours are associated with higher scores on average, but not for every individual. Choice B best evaluates the claim by noting it is too strong, as the association does not guarantee outcomes and variability exists. Choices A and C overclaim by treating association as proof of causation or guarantees, while D denies the evident relationship. A key distractor like A mistakes correlation for a fixed causal effect. When assessing scatterplots, focus on trends and spread to differentiate general patterns from absolute predictions.
A teacher wants to estimate the proportion of students in a district who own a calculator. She surveys 30 students from her own class and finds 28 own one. Which statement best describes the problem with using this result for the entire district?
The sample may be biased because students from one class may not represent the entire district, so the estimate could be inaccurate.
The sample is unbiased because 30 is large enough to represent any population automatically.
The estimate must be exact because 28 out of 30 is a precise fraction.
The estimate is invalid because ownership cannot be measured with a yes/no question.
Explanation
This question describes the issue with estimating district-wide calculator ownership from one class's survey. The data from 30 students show 28 owning one. Choice A is best, highlighting non-representative sampling from one class. Choice B assumes size alone ensures representation. Choice C mistakes estimates for exactness. For convenience samples, note they may not reflect larger populations due to bias.
A researcher reports that in a sample of 70 adults, those who own pets report higher happiness scores. The researcher concludes, “Owning a pet increases happiness.” Which statement best evaluates this conclusion?
The study shows pet ownership is associated with higher happiness in this sample, but without random assignment other factors could explain the difference, so causation is not proven.
The conclusion is proven because happiness scores are numerical and can establish cause-and-effect relationships.
The conclusion is proven because 70 adults is a large enough sample to eliminate confounding variables.
The conclusion is proven because higher happiness must cause people to buy pets, which confirms the direction of causation.
Explanation
This question assesses a researcher's conclusion that owning a pet increases happiness based on higher happiness scores among pet owners in a sample of 70 adults. The data demonstrate an association between pet ownership and higher reported happiness in this group. Choice A is best because without random assignment, factors like personality or lifestyle could confound the relationship, so causation is not established. Choice B oversteps by claiming numerical scores prove causation, which overlooks the observational study design. Choice C assumes sample size alone eliminates confounders, but size does not substitute for experimental control. Distinguish between what the data correlate and what they causally prove, especially in non-randomized studies.
A principal compares attendance rates before and after installing new hallway posters encouraging attendance. Attendance rose from 93% to 95%. No other data were collected, and the change happened during a month with fewer illnesses. Which conclusion is most appropriate?
The posters caused the attendance increase because the increase occurred after installation.
The attendance increase could be related to the posters, but seasonal illness changes or other factors could also explain it.
The posters had no effect because the increase was only 2 percentage points.
The data prove that fewer illnesses were caused by the posters since attendance increased.
Explanation
This question seeks the most appropriate conclusion from attendance data showing an increase from 93% to 95% after installing posters, during a period with fewer illnesses. The data indicate a small increase coinciding with the posters, but no controls for other factors like seasonal changes. Choice B is supported as it recognizes the possible influence of posters while noting confounding variables like illness rates that could explain the change. Choice A oversteps by claiming causation solely based on timing, without evidence isolating the posters' effect. Choices C and D dismiss or overclaim effects without basis in the data. A useful strategy is to identify potential confounders when assessing before-and-after comparisons without controls.
A local newspaper reports that neighborhoods with more parks have lower crime rates, based on citywide data collected in the same year. The paper concludes, “Building more parks will reduce crime.” Which evaluation is most appropriate?
The conclusion is justified because citywide data always allow causal conclusions.
The conclusion is not necessarily justified because the data show an association, and other factors like income or policing could affect both parks and crime.
The conclusion is justified because lower crime rates must cause cities to build more parks.
The conclusion is unjustified because crime rates cannot be compared across neighborhoods.
Explanation
The question evaluates a causal conclusion from citywide data associating more parks with lower crime. The data show a negative association across neighborhoods in one year. Choice A is appropriate, noting observational data allow association but not causation due to confounders. Choice B assumes data type guarantees causation. Choice C reverses causation. For cross-sectional data, emphasize associations versus proven causes.
A teacher notices that students who submit homework earlier tend to score higher on quizzes. The teacher says, “Submitting homework early improves quiz performance.” Which statement is most appropriate?
The statement is invalid because homework submission time cannot be compared across students.
The pattern suggests an association, but students who submit early may differ in organization or study time, so the data do not prove early submission causes higher quiz scores.
The statement is proven because earlier submission occurs before quizzes, so it must cause higher quiz scores.
The statement is proven because quiz scores cannot influence homework submission timing.
Explanation
This question evaluates a teacher's statement that submitting homework early improves quiz performance, based on noticing that early submitters tend to score higher. The data show an association between earlier submission and higher quiz scores among students. Choice A is most appropriate because confounding factors like organization or study habits could explain the pattern, so causation is not proven. Choice B oversteps by using timing to assume causation, which does not account for other variables in this observational context. Choice C assumes directionality without evidence, but the data do not confirm causation in either direction. Always differentiate observed patterns from causal claims, particularly when self-selection or unmeasured factors are possible.