Inferences & Claims From Statistics
Help Questions
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?
High light will increase germination rates for all plant species in all climates because it worked for these seeds.
The results show only correlation because random assignment prevents any conclusions about cause.
Higher light exposure likely caused a higher germination rate in this experiment because the light condition was assigned and other conditions were held constant.
Seeds that were already more likely to germinate were assigned to high light, which explains the result.
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 nutrition blogger recruited 60 volunteers from their social media followers to test a new smoothie for 4 weeks. Participants chose whether to drink the smoothie daily or not; no random assignment was used. At the end, those who drank the smoothie reported an average weight change of $-1.8$ lb, and those who did not reported $-0.4$ lb. The blogger claims the smoothie causes weight loss for adults. Which statement is justified?
The smoothie caused weight loss, since the groups were compared over the same 4-week period.
The data show that drinking smoothies is the only factor that affects weight change, because both groups were measured at the end of 4 weeks.
Adults who drank the smoothie in this volunteer sample reported greater average weight loss than those who did not, but the study design cannot show the smoothie caused the difference.
Because the smoothie group lost more weight, the smoothie will cause every adult to lose about 1.8 lb in 4 weeks.
Explanation
This question evaluates which statement is justified from a non-randomized study where volunteers chose to drink a smoothie and reported weight changes. The data indicate that the smoothie group reported greater average weight loss (-1.8 lb vs. -0.4 lb) over 4 weeks. Choice A is supported because it acknowledges this difference in the sample but notes the lack of random assignment prevents establishing causation. Choices B, C, and D overstep by claiming universal causation or that the smoothie is the sole factor, ignoring self-selection bias. For instance, B generalizes to all adults without evidence, overreaching the volunteer sample. A useful strategy is to check for experimental design elements like randomization before accepting causal claims from group differences.
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.
The decrease on 5 routes proves the schedule is effective, since comparing to the previous month eliminates all confounding variables.
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.
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 researcher surveyed 300 randomly selected adults in one state about commuting time and found an average of 27 minutes. She concludes, “Adults in the United States commute about 27 minutes on average.” Which statement best evaluates this conclusion?
The conclusion is correct because 300 is large enough to represent any population in any location.
The conclusion is correct because random sampling in one state automatically represents all states.
The conclusion is incorrect because commuting time cannot be averaged across people.
The conclusion may be an overgeneralization because the sample represents adults in one state, not necessarily the entire United States.
Explanation
The question evaluates generalizing a state sample's commuting average to the entire U.S. The data from 300 random adults in one state yield 27 minutes. Choice A is best, noting overgeneralization as one state may not represent national variations. Choice B wrongly assumes state samples represent nations. Choice C ignores representation issues. For geographic generalizations, ensure the sample's scope matches the claimed population.
A principal wants to estimate the percentage of students at a large high school who usually ride the bus. She surveys 80 students who arrive early for a morning club meeting and finds that 52% report usually riding the bus. Which statement best describes whether this result can be used to estimate the bus-riding percentage for the entire school?
It is biased only if fewer than half of the surveyed students ride the bus, which is not the case here.
It is unbiased because surveying any students at the school automatically produces a random sample of the whole school population.
It is unbiased because 80 students is a sufficiently large sample to represent the entire school regardless of how they were selected.
It is likely biased because students who arrive early for a club meeting may differ from the overall student body in transportation habits.
Explanation
This question addresses sampling bias when estimating school-wide bus ridership. The principal surveyed students arriving early for a club meeting - not a random sample of all students. Choice A correctly identifies this as "likely biased because students who arrive early for a club meeting may differ from the overall student body in transportation habits" - club members might live closer, have parent drivers, or differ in other ways affecting transportation. Choice B wrongly claims any 80 students would be unbiased regardless of selection method. For unbiased estimates, samples must be randomly selected from the population of interest.
A teacher wants to know whether a new review worksheet improves quiz scores. Two existing classes were used: Class 1 (22 students) chose to use the worksheet; Class 2 (24 students) did not. Both classes took the same quiz. The average score in Class 1 was 84, and in Class 2 it was 78. Which limitation most affects any claim that the worksheet caused the higher average score?
Because both classes took the same quiz, the worksheet must be the only reason the averages differed between classes.
Students were not randomly assigned to use the worksheet, so differences between the two classes could explain the score difference.
Since the average score increased by 6 points, every student who used the worksheet scored exactly 6 points higher than they otherwise would have.
The sample size is over 40 students total, so the worksheet’s effect can be generalized to all students in the school district.
Explanation
This question asks about limitations in comparing two classes where students self-selected whether to use a worksheet. The key issue is that students were not randomly assigned - Class 1 chose to use the worksheet while Class 2 did not. Choice A correctly identifies this limitation: without random assignment, the classes might differ in motivation, prior knowledge, or other factors that could explain the score difference. Choice C incorrectly assumes the worksheet "must be the only reason" when self-selection creates confounding variables. In educational research, random assignment is crucial for establishing causal effects.
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.)
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.
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.
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 random sample of 80 students at a large university reported how many hours per week they work for pay. The sample median was 12 hours. Which statement is most accurate?
The sample median of 12 hours is an estimate of the university’s true median work hours, but the population median could differ due to sampling variability.
The population median must be exactly 12 hours because medians from random samples always match the population median.
The result proves that exactly half of all university students work 12 hours per week.
The median cannot be used because only means can summarize hours worked.
Explanation
This question asks which statement is most accurate for a sample median of 12 work hours from 80 students. The data provide an estimate, but not the exact population value. Choice A is supported as it recognizes sampling variability. Choice B oversteps by claiming exact match. Choices C and D dismiss medians or overclaim. Samples estimate but do not equal population parameters.
A study followed 500 adults for one year and found that people who reported eating breakfast at least 5 days per week had a lower average BMI than those who ate breakfast fewer days. The study did not assign breakfast habits. Which statement is best supported?
Eating breakfast at least 5 days per week causes a lower BMI for all adults, so skipping breakfast increases BMI.
In this group, frequent breakfast eaters had lower average BMI, but other factors such as exercise or income could explain the difference.
The results prove that BMI causes people to eat breakfast more often.
Because the sample size is 500, the results cannot be influenced by confounding variables.
Explanation
The question finds the best-supported statement from observational data on breakfast frequency and BMI among adults. The data show lower average BMI for frequent breakfast eaters over a year. Choice B is appropriate, noting the association but possible confounders like exercise. Choice A claims causation universally. Choice C reverses causation. In longitudinal observations, data support associations but not causation without controls.
A teacher randomly selects 40 essays from all essays submitted and finds the mean score is 86. The teacher states, “The class mean is 86.” Which statement is most accurate?
The statement is guaranteed because random samples always produce the exact population mean.
The statement is guaranteed because 40 is an even number and even samples match populations.
The statement may be inaccurate because 86 is the sample mean, and the true class mean could differ unless all essays were included.
The statement is invalid because essay scores cannot be averaged.
Explanation
This question assesses a teacher's statement that the class mean score is 86 based on a random sample of 40 essays. The data provide a sample mean of 86, but not the full population. Choice A is supported as it notes the sample mean estimates but does not guarantee the population mean due to variability. Choice B oversteps by claiming random samples always match exactly, ignoring sampling error. Choices C and D misuse sampling concepts or dismiss averaging. Remember, sample statistics estimate population parameters but are not identical.