Inferences & Claims From Statistics
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SAT Math › Inferences & Claims From Statistics
A school surveyed 120 students who voluntarily visited the library last week and recorded their average nightly sleep and their math test score (out of 100) from the same week. The results showed a positive association: students reporting more sleep tended to have higher scores. The counselor claims, “If any student sleeps 9 hours per night, their math score will increase.” Which conclusion is most appropriate based on this observational survey?
Because the sample size is 120, the results must generalize to all students in the country, regardless of differences in schools or schedules.
The data show an association in this volunteer sample, but they do not prove that increasing sleep will raise an individual student’s math score.
The data prove that students who sleep less are not studying enough, since lower sleep was linked with lower scores in the survey.
Sleeping 9 hours per night causes higher math scores for all students, because the survey found a positive association between sleep and scores in the sample.
Explanation
This question asks which conclusion is most appropriate from an observational survey showing a positive association between sleep and math scores in a volunteer sample of 120 students. The data indicate that, in this group, students reporting more sleep tended to have higher scores, but the study was not experimental and relied on self-selected participants. Choice B is supported because it accurately recognizes the association without claiming causation or overgeneralizing, emphasizing that the results do not prove increasing sleep will raise scores for any individual. In contrast, choice A oversteps the evidence by asserting causation, ignoring potential confounders like study habits that could explain both more sleep and higher scores. Similarly, choice D incorrectly assumes the small volunteer sample represents all students nationwide. When evaluating claims from surveys, always distinguish between observed associations in a specific sample and unproven causal effects or broad generalizations.
A researcher analyzed data from 15 countries and found that countries with higher average income tended to have longer life expectancy. The researcher claims, “Raising a country’s income will increase life expectancy.” Which statement is most justified?
The data prove the same relationship must hold for every individual person within each country, because country averages equal individual effects.
The data support a positive association across these countries, but they do not prove that increasing income would cause life expectancy to rise.
The data prove that increasing income is the only cause of longer life expectancy, because the pattern holds across multiple countries.
The data prove that longer life expectancy causes higher income, because causation must run from health outcomes to economic outcomes.
Explanation
This question evaluates a researcher's claim that raising income will increase life expectancy, based on data from 15 countries showing higher income linked to longer expectancy. The data indicate a positive association across countries, but it's observational. Choice A is most justified as it reports the association without proving causation. Choice B oversteps by claiming income is the only cause. Choice D wrongly equates country averages to individual effects. A key strategy is to distinguish cross-sectional associations from causal proof, considering other national factors.
A scientist tested a fertilizer on 30 identical plants by randomly assigning 15 to receive Fertilizer X and 15 to receive no fertilizer, keeping light and water the same. After 4 weeks, mean height was 18.4 cm with Fertilizer X and 15.9 cm with no fertilizer. Which conclusion is most appropriate?
Fertilizer X likely caused greater average plant height under these conditions, since random assignment and a control group support causation.
The study shows only correlation because plant growth always depends on many variables, so experiments cannot support causal claims.
The study proves Fertilizer X increases height for all plant species in all environments, because the experiment used random assignment.
Fertilizer X will make every plant exactly 2.5 cm taller, because the difference in group means must apply to each individual plant.
Explanation
The question asks for the most appropriate conclusion from a randomized experiment where Fertilizer X led to taller average plant height (18.4 cm) than no fertilizer (15.9 cm). The data show a difference in means, with random assignment and a control group. Choice A is supported as it infers causation under the study's conditions. Choice B oversteps by claiming exact individual effects. Choice D wrongly overgeneralizes to all species and environments. When interpreting experiments, limit causal claims to the tested setup, distinguishing group averages from individual guarantees.
A study examined 90 middle school students and found that those who ate breakfast had higher average attention ratings during first period than those who did not. Breakfast choice was not assigned; students chose whether to eat. Which statement is best supported by the study?
Breakfast has no relationship to attention, because self-reported choices make it impossible for any association to appear in data.
Students who ate breakfast had higher average attention ratings in this sample, but other factors like sleep or home routines could explain the difference.
Not eating breakfast causes low attention for all students, because the study included 90 students and therefore represents every middle schooler.
Eating breakfast causes higher attention in first period, because higher average attention among breakfast-eaters proves a causal relationship.
Explanation
This question determines what is best supported by a study where breakfast-eaters had higher average attention ratings than non-eaters among 90 middle schoolers, with self-chosen breakfast. The data show an association in the sample, but it's observational. Choice B is justified as it notes the difference while considering confounders like sleep. Choice A oversteps by claiming causation from the association. Choice C wrongly generalizes to all students. A strategy is to remember self-reported behaviors in observational studies suggest links but not causation, due to possible external factors.
A researcher sampled 50 households from a small town and found the mean monthly electricity bill was $\$112$. The researcher concludes, “The mean monthly electricity bill for all households in the state is $$112$.” Which statement is most appropriate?
The conclusion is justified because the town is in the state, so the town’s mean must equal the state’s mean by definition.
The conclusion may be an overgeneralization because the sample is from one small town and may not represent households statewide.
The conclusion is justified because electricity bills do not vary by location, so any town’s mean equals the state mean.
The conclusion is justified because a sample mean always equals the population mean when the sample size is at least 50.
Explanation
The question assesses a researcher's conclusion that the state mean electricity bill is $112, based on a sample mean from 50 households in one small town. The data provide an estimate from this town, but it's not randomly selected statewide. Choice B is most appropriate as it highlights overgeneralization from a localized, non-representative sample. Choice A oversteps by assuming sample size ensures equality to the population. Choice C incorrectly assumes no variation by location. When making population inferences, ensure the sample is representative of the broader group.
A teacher wants to know whether a new seating arrangement reduces classroom disruptions. For 2 weeks, the teacher counts disruptions with the old arrangement (average 9 per day). Then for the next 2 weeks, the teacher uses the new arrangement (average 6 per day). The teacher claims the new arrangement caused fewer disruptions. Which is the best evaluation?
The claim is proven because 2 weeks is long enough to eliminate all confounding variables in a classroom setting.
The claim is proven because disruptions are counts, so the only possible explanation for a decrease is the new arrangement.
The claim is supported because the average decreased, and any before-and-after change must be caused by the seating arrangement alone.
The claim is weakened because other changes over time, such as different lessons or student behavior trends, could explain the decrease.
Explanation
This question evaluates a teacher's claim that a new seating arrangement caused fewer disruptions, based on averages dropping from 9 to 6 per day over two-week periods. The data show a decrease after the change, but it's a before-and-after comparison without controls. Choice B is appropriate as it notes potential confounders like lesson changes over time. Choice A oversteps by assuming the change must be causal. Choice C wrongly claims the duration eliminates confounders. A key strategy is to recognize uncontrolled time-series data suggest but do not prove causation due to external influences.
A political poll interviewed 600 adults using landline phone numbers listed in a directory and found that 55% support Candidate X. The pollster claims, “55% of all adults in the state support Candidate X.” Which concern most directly affects whether the claim is justified?
The claim is automatically correct because directory-listed landlines guarantee random selection of all adults in the state.
The sample may be unrepresentative because adults without listed landlines could be undercovered, leading to coverage bias in the estimate.
The claim must be wrong because no candidate can ever receive support from more than 50% of adults in a state.
The sample is automatically representative because 600 is a large number, so the method of selecting respondents does not matter.
Explanation
The question concerns whether a poll's claim of 55% support for a candidate among all state adults is justified from 600 landline interviews. The data show 55% support in this sample, but the method uses listed landlines only. Choice A is best as it identifies coverage bias from excluding those without landlines. Choice B oversteps by assuming size alone ensures representation. Choice D incorrectly claims the method is random. When evaluating polls, consider if the sampling frame covers the entire population to avoid bias.
A gym posts a sign: “Members who attend 4+ classes per week lose more weight.” The gym’s records show that, over 3 months, members attending 4+ classes lost an average of 6 lb, while others lost an average of 2 lb. Attendance was self-chosen. Which statement is best supported?
Attending 4+ classes per week caused the extra weight loss, because the gym compared two groups over the same 3-month period.
The records prove that class attendance cannot be related to weight loss, because weight loss depends only on diet and never on exercise.
The records show an association between higher class attendance and greater average weight loss, but motivated members may differ in other ways.
The records prove that every member who attends fewer than 4 classes per week will lose exactly 2 lb over 3 months.
Explanation
This question assesses what is best supported by gym records showing higher average weight loss (6 lb) for members attending 4+ classes per week versus 2 lb for others, with self-chosen attendance. The data indicate an association between attendance and weight loss, but it's observational. Choice B is justified as it reports the association while noting potential confounders like motivation. Choice A oversteps by claiming causation without randomization. Choice C wrongly assumes exact individual outcomes from averages. A strategy is to recognize self-selection can introduce biases, distinguishing observed patterns from proven causes.
A researcher measured the relationship between hours studied and exam score for 35 students in an advanced placement class and found a strong positive correlation. The researcher concludes, “Studying more will increase exam scores for all high school students.” Which critique is most appropriate?
The conclusion is incorrect because positive correlations cannot occur between study time and exam scores in real data.
The conclusion is correct because a strong correlation always implies causation, regardless of how the data were collected.
The conclusion overgeneralizes, since the sample is from one advanced class and may not represent all high school students.
The conclusion is correct because 35 students is enough to guarantee the same relationship holds for every student nationwide.
Explanation
The question critiques a researcher's conclusion that more studying increases exam scores for all high school students, based on a strong positive correlation in 35 advanced placement students. The data show a correlation in this specific class, but the sample is small and selective. Choice A is most appropriate as it addresses overgeneralization to all students from one advanced group. Choice B oversteps by assuming correlation always implies causation. Choice C incorrectly claims the sample size guarantees universality. When evaluating broad claims, check if the sample's selectivity limits inferences to wider populations.
A study tracked 500 adults for one year and found that people who took a daily multivitamin reported fewer sick days than those who did not. Participants chose whether to take vitamins; there was no random assignment. Which statement is most justified?
The study proves that multivitamins have no effect, because self-selection invalidates all numerical comparisons between groups.
The study proves that everyone should take a multivitamin, because the sample size is large enough to guarantee identical effects for all adults.
The study shows an association, but healthier lifestyles or other differences between groups could explain the fewer sick days.
Taking a multivitamin caused fewer sick days, because following people over time always establishes causation even without random assignment.
Explanation
This question identifies the most justified statement from an observational study where self-selected vitamin users reported fewer sick days than non-users over one year. The data show an association with fewer sick days among vitamin takers, but without randomization. Choice B is appropriate as it notes the association while acknowledging possible confounders like overall health habits. Choice A oversteps by claiming causation from a longitudinal but non-experimental design. Choice D wrongly generalizes to all adults. A useful strategy is to differentiate associations in self-selected groups from causal evidence, considering lifestyle differences.