Inferences & Claims From Statistics
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SAT Math › Inferences & Claims From Statistics
A principal claims that a new attendance policy reduced absences. The school compares the average number of absences per student in the semester before the policy (3.1) to the semester after (2.7). During the after semester, a flu outbreak was milder and several students transferred out. Which conclusion is most appropriate?
The decrease in average absences is consistent with the policy helping, but other changes between semesters could explain the difference, so causation is uncertain.
The policy definitely caused the reduction in absences because the average absences decreased after the policy was introduced.
The policy had no effect because averages can never be used to compare two semesters.
The policy reduced absences for every student by exactly 0.4 absences, because the average decreased by 0.4.
Explanation
This question examines whether a new attendance policy caused reduced absences when comparing semesters. Average absences dropped from 3.1 to 2.7, but the second semester also had a milder flu outbreak and student transfers. Choice B correctly recognizes that while the decrease is consistent with the policy helping, other changes between semesters could explain the difference, making causation uncertain. Choice A incorrectly assumes the policy definitely caused the change. Choice C dismisses comparison inappropriately. Choice D claims unrealistic individual-level effects from an average. When multiple factors change between time periods, you cannot isolate the effect of any single change.
A nutritionist wants to compare two diets on cholesterol levels. She randomly assigns 100 volunteers to Diet A and 100 volunteers to Diet B for 3 months and measures the mean change in LDL cholesterol. Diet A’s mean change is $-12$ mg/dL and Diet B’s is $-5$ mg/dL. Which statement is most appropriate?
Diet B is better than Diet A for all populations, because Diet B was tested on the same number of volunteers as Diet A.
Diet A likely caused a larger average reduction in LDL than Diet B for these volunteers, because random assignment supports a causal interpretation.
Diet A will reduce every person’s LDL by exactly 12 mg/dL, because the mean change for Diet A is $-12$ mg/dL.
Diet A is associated with lower LDL, but random assignment prevents concluding Diet A caused the reduction.
Explanation
This question presents results from a randomized experiment comparing two diets' effects on LDL cholesterol. With 100 volunteers randomly assigned to each diet, Diet A showed a mean reduction of 12 mg/dL while Diet B showed 5 mg/dL. Choice A correctly concludes that Diet A likely caused a larger reduction for these volunteers, which is appropriate given random assignment in an experiment. Choice B incorrectly suggests random assignment prevents causal conclusions when it actually enables them. Choice C unrealistically claims identical effects for every person. Choice D makes inappropriate population generalizations and ignores that A performed better. Randomized experiments support causal conclusions about average treatment effects.
A student council polled 80 students who were in the cafeteria at lunch and found that 60% supported extending lunch by 5 minutes. The council wants to claim that 60% of all students in the school support the change. Which statement best evaluates this claim?
The claim must be false because a sample proportion can never equal a population proportion.
The claim is guaranteed correct because 80 is a sufficiently large sample for any school-wide conclusion.
The claim proves that extending lunch will increase academic performance, because more students support the change than oppose it.
The claim may be unreliable because students in the cafeteria at lunch may not be representative of all students in the school.
Explanation
This question examines whether a cafeteria lunch sample can represent all students in a school. The student council surveyed only students who were in the cafeteria at lunch, which is a convenience sample that may not represent all students. Choice A correctly identifies this sampling bias - students who eat elsewhere, have different lunch periods, or skip lunch entirely are excluded, potentially biasing results. Choice B incorrectly claims sample size alone ensures representativeness. Choice C inappropriately introduces causation to a descriptive poll. Choice D makes a false claim about sample and population proportions. The location and timing of sampling can create systematic bias regardless of sample size.
A meteorologist recorded the number of ice cream sales and the daily high temperature for 30 days and found a strong positive correlation. Which statement is most appropriate?
Higher temperatures cause people to buy ice cream, because correlation always implies causation when the correlation is strong.
Ice cream sales and temperature are associated in the data, and a third factor such as season or weather conditions could explain the relationship.
Higher ice cream sales cause higher temperatures, because ice cream sales and temperature are strongly positively correlated.
There is no relationship between ice cream sales and temperature because the data were collected for only 30 days.
Explanation
This question examines the classic correlation between ice cream sales and temperature. The data shows a strong positive correlation over 30 days of observation. Choice C correctly identifies the association while noting that a third factor (season/weather) could explain why both variables move together - hot weather increases both temperature readings and ice cream demand. Choice A illogically suggests ice cream sales cause temperature. Choice B incorrectly states correlation always implies causation. Choice D dismisses a clear relationship based on sample size. This is a textbook example of how correlation doesn't imply causation, especially when an obvious third variable (weather) affects both measured variables.
A researcher wants to know whether a new tutoring program improves algebra scores. She lets students choose whether to join tutoring and then compares end-of-term scores between participants and nonparticipants. Participants scored higher on average. Which statement best evaluates the claim that tutoring caused the increase?
Because algebra scores were measured at the end of the term, the study can be generalized to all subjects, including history and biology.
Because participants scored higher, the tutoring program must have caused the score increase for every student who joined.
Because the researcher compared two groups, the study is automatically a randomized experiment and therefore proves tutoring caused higher scores.
Because students self-selected into tutoring, differences in motivation or prior achievement could explain the higher scores, so causation is not established.
Explanation
This question examines whether a tutoring program caused higher algebra scores when students self-selected into the program. The key issue is self-selection bias - students who chose tutoring likely differed in motivation, prior achievement, or other factors from those who didn't. Choice A correctly identifies that these pre-existing differences could explain the higher scores, making causation uncertain despite the observed difference. Choice B incorrectly assumes causation from self-selected groups. Choice C falsely claims any two-group comparison is experimental. Choice D makes an inappropriate generalization to other subjects. Without random assignment, differences between self-selected groups cannot establish causation.
A wildlife biologist captured and tagged 50 turtles in one lake and recorded their shell lengths. From this sample, the biologist estimated the mean shell length of turtles in that lake. Which statement correctly distinguishes the sample from the population?
The sample is all turtles in the lake, and the population is the 50 tagged turtles used to compute the mean.
The sample is the 50 tagged turtles, and the population is all turtles in the lake that the biologist wants to describe.
The population is all turtles everywhere, because any turtle study automatically generalizes to turtles worldwide.
Both the sample and the population are only the 50 tagged turtles because statistics cannot be used to describe unobserved turtles.
Explanation
This question tests understanding of sample versus population in statistical inference. The biologist measured 50 specific turtles (the sample) to make inferences about all turtles in the lake (the population). Choice B correctly identifies the 50 tagged turtles as the sample and all turtles in the lake as the population of interest. Choice A reverses these definitions. Choice C incorrectly limits both to only observed turtles. Choice D inappropriately extends the population to all turtles everywhere when the study focused on one specific lake. Remember: the sample is what you actually measure, while the population is the larger group you want to learn about.
A cafe tested whether changing background music affects how long customers stay. For one week, it played slow music; for the next week, it played fast music. The average time stayed increased from 18 minutes to 22 minutes. No other variables were controlled, and the weeks had different weather and promotions. Which statement is best supported?
The change in average stay time could be due to music or to other differences between the two weeks, so the data do not justify a causal claim about music.
The results must generalize to all cafes because the cafe observed real customers rather than a laboratory sample.
Fast music caused customers to stay longer because the average time increased during the fast-music week.
Slow music always reduces customer stay time by exactly 4 minutes, because the difference between the two weekly averages is 4 minutes.
Explanation
This question describes a cafe comparing customer stay times between two different weeks with different music. The average increased from 18 to 22 minutes when switching from slow to fast music, but the weeks also had different weather and promotions. Choice B correctly recognizes that without controlling other variables, we cannot determine whether music or other differences between weeks caused the change in stay time. Choice A incorrectly attributes the change solely to music. Choice C makes unrealistic claims about exact effects. Choice D inappropriately generalizes to all cafes. When conditions change between comparison periods, multiple factors could explain observed differences.
A health blog claims that “people who drink more water have fewer headaches.” The blog cites an observational study of 200 adults that found a correlation between daily water intake and weekly headache frequency. The study did not measure caffeine intake, stress, or sleep. Which statement is most appropriate?
The study suggests an association between water intake and headache frequency, but confounding variables could explain the relationship, so causation is not established.
The study proves that stress cannot affect headaches, because the study focused on water intake instead of stress.
The study implies that all adults should drink the same amount of water to eliminate headaches entirely.
The study shows drinking more water causes fewer headaches, because a correlation was found in a sample of 200 adults.
Explanation
This question examines a health claim based on an observational study showing correlation between water intake and headache frequency. The study found an association but didn't measure potential confounding variables like caffeine, stress, or sleep that could affect both water intake and headaches. Choice B correctly notes the association while identifying that confounding variables could explain the relationship, so causation is not established. Choice A incorrectly claims the correlation proves causation. Choice C makes an illogical claim about stress. Choice D inappropriately suggests universal water recommendations. When evaluating health claims from observational studies, look for unmeasured confounding factors that could create spurious correlations.
A researcher wants to estimate the proportion of all voters in a state who support a new transit tax. She surveys 500 people by calling landline phone numbers listed in a directory. Which statement best describes a concern about generalizing the results to all state voters?
The sample must be unbiased because 500 is a large number, so it will always represent the state population well.
The survey will prove whether the transit tax causes people to vote more often because it asks about support for the tax.
Generalization is guaranteed because phone surveys always reach a random sample of all voters in the state.
The sample may be biased because voters without listed landlines are less likely to be included, so the results may not represent all voters in the state.
Explanation
This question asks about concerns when generalizing from a landline phone survey to all state voters. The key issue is sampling bias - the researcher only called landline numbers from a directory, systematically excluding voters without listed landlines. Choice A correctly identifies this bias and explains why results may not represent all voters, as younger voters, cell-phone-only households, and unlisted numbers are excluded. Choice B incorrectly claims large samples guarantee unbiased results. Choice C inappropriately brings up causation for a descriptive survey. Choice D falsely claims phone surveys always reach random samples. Remember that how you select your sample matters more than how many you survey - biased sampling methods produce biased results regardless of sample size.
A gym tracked 10 members for one month and recorded each member’s number of weekly visits and their change in body mass (kg). The line of best fit on the graph slopes downward. Which conclusion is most appropriate from these data?
Every additional gym visit per week guarantees the same amount of weight loss for all members, since the data show a trend.
The gym can conclude there is no relationship because only 10 members were tracked and a relationship requires at least 30 people.
More gym visits caused members to lose weight, because the best-fit line slopes downward.
Members who visited the gym more often tended to have more negative mass changes, but the data do not prove that visits caused the change.
Explanation
This question presents data showing a downward-sloping best-fit line between gym visits and body mass change for 10 members. The negative slope indicates that members with more visits tended to have more negative mass changes (weight loss). Choice B correctly describes this association while noting that the observational data cannot prove visits caused the weight change - members who visit more might differ in diet, motivation, or other factors. Choice A incorrectly claims causation from correlation. Choice C unrealistically claims every visit guarantees identical effects for all people. Choice D incorrectly dismisses the relationship based on sample size. When you see a trend in observational data, describe the association but don't assume causation.