- A principal wants to estimate the proportion of students at her school who bike to school. She surveys 80 students by standing near the bike racks before first period and asking every 5th student who passes. Of the 80 surveyed, 44 said they bike. Which statement about the estimate is most appropriate?
Answer: The estimate 44/80 is likely biased upward because the sampling method over-represents students who are near the bike racks.
This question evaluates an estimate of biking proportion from a non-random survey near bike racks. The data shows 44 out of 80 surveyed students bike, estimating 55%. Choice A is most appropriate because the sampling method biases toward bikers by location, likely overestimating the proportion. Choice B incorrectly claims unbiasedness, as selecting every 5th passerby does not ensure randomness for the whole school. Choice C oversteps by treating the estimate as exact proof, ignoring sampling bias. In sampling questions, identify biases from non-random methods that can skew estimates away from the true population value. - A study of 200 college students found a positive correlation between the number of hours spent studying and GPA. Which conclusion is most appropriate?
Answer: There is an association between studying hours and GPA, but causation cannot be determined.
This question asks what conclusion is most appropriate from a positive correlation between study hours and GPA. The data shows that students who study more hours tend to have higher GPAs, indicating a positive association between these variables. Answer D correctly identifies that while there is an association, causation cannot be determined from correlational data alone. Options A and C incorrectly claim causation (that studying more causes higher GPA), while option B suggests reverse causation, but none of these causal claims can be supported by correlation data. When dealing with correlational studies, always remember that correlation does not imply causation.
- A researcher wants to know whether listening to music while reading affects comprehension. She randomly assigns 90 students to one of three conditions: no music, instrumental music, or music with lyrics. After reading the same passage, students take the same comprehension test. Mean scores are 84 (no music), 86 (instrumental), and 79 (lyrics). Which statement is most appropriate?
Answer: For these students, the presence and type of music can be considered a cause of differences in average comprehension scores, because the conditions were randomly assigned.
This question involves a randomized experiment testing music's effect on reading comprehension. With 90 students randomly assigned to three conditions (no music, instrumental, lyrics), the design allows causal conclusions about how music presence and type affected average comprehension scores. The results show instrumental music had the highest mean (86), no music intermediate (84), and lyrics lowest (79), differences that can be attributed to the conditions due to randomization. Choice A correctly states that music conditions can be considered a cause of score differences for these students. Choice B overgeneralizes beyond the study participants and context.
- A sports scientist recorded the number of hours per week 30 athletes spent strength training and their sprint times (seconds) for a 100-meter dash. Athletes were not assigned training hours; they followed their own routines. The scientist found that athletes who trained more tended to have lower sprint times. Which conclusion is most appropriate?
Answer: In this group of athletes, more strength training was associated with faster sprint times, but other factors like coaching or prior fitness could explain the pattern.
This question examines an observational study where athletes' self-chosen strength training hours were compared to their sprint times. The scientist found that athletes training more hours tended to have faster (lower) sprint times, indicating an association. However, since training hours weren't randomly assigned, this relationship could be explained by factors like natural athletic ability, coaching quality, or dedication - athletes who train more might also have other advantages. Choice B correctly describes the association while noting that other factors could explain the pattern. Choice A incorrectly claims causation from observational data, ignoring potential confounding variables.
- A teacher compared two existing class periods and found that Period 1 (25 students) had an average of 7.8 hours of sleep and an average test score of 84, while Period 2 (27 students) had 6.6 hours of sleep and an average test score of 78. Students were not randomly assigned to class periods. Which limitation most affects a causal conclusion about sleep and test scores?
Answer: The teacher did not randomly assign students to different sleep amounts, so other differences between periods could explain the score difference.
The question asks which limitation most undermines a causal claim linking sleep to test scores. Because the teacher compared two intact class periods rather than assigning students to sleep amounts, the data are observational, so the real concern is that other differences between the periods, such as time of day, course load, who enrolled, or motivation, could produce the 84-versus-78 score gap instead of the 1.2-hour sleep difference. The statement that studying students makes results automatically generalize to all teenagers in the country is backwards: two classes taught by one teacher are a narrow, non-random sample, and generalizing is a separate issue from causation. Using averages instead of medians does not make comparison impossible, since means are a standard summary here. And unequal sample sizes, 25 versus 27, neither block comparison nor force population scores to match.
- 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?
Answer: 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.
The question asks what threatens generalizing a 500-person survey to all state voters. The selection method is the problem: calling only landline numbers listed in a directory means entire groups, including cell-phone-only households, unlisted numbers, and younger voters, had no chance of being reached, so the sample can differ systematically from the electorate. That is exactly the concern raised by the statement about voters without listed landlines being less likely to be included. The response claiming 500 is large enough to guarantee representativeness confuses size with fairness; a big sample drawn badly is just a precisely measured biased estimate. The claim that phone surveys always reach a random sample is false, since the directory decides who can be contacted. And the statement about proving the tax causes higher turnout invents a causal question the survey never asked.
- 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?
Answer: 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.
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 nutrition study randomly assigned 150 adults to one of three diets for 8 weeks: Diet A, Diet B, or Diet C (50 per group). The line graph shows average weight change (lb) over time; Diet B shows the largest average loss by week 8. Which inference is most appropriate?
Answer: Diet B caused greater average weight loss than Diet A and Diet C for these participants, because the diets were randomly assigned and Diet B's average loss was greatest by week 8.
This question involves a randomized nutrition study comparing three diets over 8 weeks. With 150 adults randomly assigned to diets (50 per group), the study design allows causal conclusions about which diet produced greater average weight loss for these participants. Diet B showing the largest average loss by week 8 can be attributed to the diet itself rather than participant characteristics, thanks to randomization. Choice A correctly states that Diet B caused greater average weight loss for these participants due to random assignment. Choice B overgeneralizes to "every person," while Choice C incorrectly claims only correlation despite the randomized design.
- Which conclusion is most appropriate based on the study's findings?
Answer: There is a negative correlation between exercise and stress levels in the sample.
This question asks which conclusion is most appropriate based on a study finding a negative correlation between exercise hours and stress levels among college students. The data shows that as exercise hours increase, stress levels tend to decrease within this specific sample. The correct answer (B) accurately describes this relationship as a negative correlation in the sample, using appropriate statistical language. Choice A incorrectly claims causation and overgeneralizes to "all college students," while choice C makes an absolute guarantee that the data cannot support. Choice D directly contradicts the stated findings. When interpreting correlational studies, focus on what the data actually demonstrates rather than inferring causation or universal applicability.
- An environmental club recorded the number of plastic bottles collected at weekly cleanups and the number of volunteers who attended. The line graph shows both quantities rising over 8 weeks. A member claims, "More volunteers caused the increase in bottles collected." Which conclusion is most appropriate from the graph alone?
Answer: The graph shows that weeks with more volunteers tended to coincide with more bottles collected, but it does not rule out other factors such as location or weather.
This question tests interpretation of a line graph showing both volunteer numbers and bottles collected rising over 8 weeks. While the graph shows these quantities tend to increase together (a positive association), it cannot establish that more volunteers caused more bottle collection. Other factors could explain both increases - perhaps growing community awareness led to both more volunteers and more littering cleanup, or seasonal factors affected both variables. The graph alone cannot rule out these alternative explanations or prove causation. Choice B incorrectly claims a fixed causal relationship, while choice A correctly notes the association while acknowledging that other factors could be involved.
- 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?
Answer: Because students self-selected into tutoring, differences in motivation or prior achievement could explain the higher scores, so causation is not established.
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 survey of 500 employees in a company found that job satisfaction is higher among those who work flexible hours. What can be concluded from the study?
Answer: There is an association between flexible hours and job satisfaction.
The question asks what a survey of 500 employees showing higher job satisfaction among those with flexible hours supports. Because the schedules were observed rather than assigned, the justified conclusion is the modest one: there is an association between flexible hours and job satisfaction. The statement that flexible hours cause higher job satisfaction goes past the evidence, since employees in more autonomous or better-paid roles may be the ones granted flexible schedules, meaning the schedule and the satisfaction could share a common source. The claim that employees prefer working flexible hours describes a preference the survey never measured; it recorded satisfaction levels, not stated choices. And the claim that all flexible-hours employees are satisfied turns a group-level tendency into a universal, which comparisons of group satisfaction cannot support.
- Researchers conducted a survey of 200 adults to determine if there is an association between daily coffee consumption and reported levels of stress. The results showed a moderate positive correlation. Which statement is justified by the data?
Answer: People who drink more coffee tend to report higher stress levels.
The question asks what a moderate positive correlation between daily coffee consumption and reported stress among 200 adults actually justifies. Positive correlation means the two measures rise together, so the supportable statement is that people who drink more coffee tend to report higher stress levels; the phrase "tend to" keeps the claim at the level of association, which is what survey data can show. Saying that drinking coffee increases stress levels asserts a direction of cause and effect the design cannot establish, since stressed people may simply reach for more coffee, or long work hours may drive both. The statement that eliminating coffee will reduce stress for all individuals goes further still, predicting an intervention's effect on every person. And the claim that there is no relationship contradicts the reported correlation outright.
- 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?
Answer: Diet A likely caused a larger average reduction in LDL than Diet B for these volunteers, because random assignment supports a causal interpretation.
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.
- Based on the study, which conclusion is justified?
Answer: Higher caffeine intake is associated with better concentration scores.
This question asks which conclusion is justified based on research showing a moderate positive correlation between caffeine intake and concentration scores. The study data reveals an association where higher caffeine consumption corresponds with better concentration test performance within the sample. Choice B correctly describes this relationship as an association, using appropriate correlational language that doesn't overstate the findings. Choice A incorrectly claims causation and universal applicability, choice C suggests caffeine is the only factor (which correlation cannot establish), and choice D contradicts the positive correlation findings. Remember that correlation studies reveal relationships between variables but cannot establish causation or guarantee individual outcomes.
- 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?
Answer: The study suggests an association between water intake and headache frequency, but confounding variables could explain the relationship, so causation is not established.
This study is observational: nobody assigned water intake, and caffeine, stress, and sleep were never measured. So the defensible statement is the one saying the data suggest an association while confounding variables could explain it, leaving causation unestablished. A stressed, sleep-deprived, coffee-heavy adult might both drink less water and get more headaches, producing the correlation without water doing anything. The statement that drinking more water causes fewer headaches treats a correlation as proof of cause, which no observational design can deliver. The statement that all adults should drink the same amount to eliminate headaches entirely goes further still, turning a group trend into a universal prescription and promising a complete cure the data never demonstrate. The statement that stress cannot affect headaches misreads the study's silence, since failing to measure a variable is not evidence that it has no effect.
- A study investigated the relationship between the number of hours students study per week and their final exam scores. The study surveyed 100 students and found a positive correlation between study hours and exam scores. Which conclusion is most appropriate based on the study?
Answer: Students who study more tend to have higher exam scores.
This question asks what conclusion is most appropriate based on a study showing a positive correlation between study hours and exam scores. The data shows that students who study more hours tend to score higher on exams, indicating an association between these variables. Choice B correctly states this relationship using appropriate language ("tend to have") that reflects correlation without implying causation.
- A researcher wants to estimate the average number of books read per year by adults in a city of 300,000 people. She surveys 200 adults by standing outside a downtown library on Saturday afternoon and asking passersby. The sample mean is 18 books per year. Which statement best evaluates using this result to estimate the citywide average?
Answer: The estimate is likely biased upward because people near a library may read more than the average adult in the city, so the sample may not represent the population.
The question asks how much a sample mean of 18 books, collected from passersby outside a downtown library on a Saturday afternoon, tells us about a city of 300,000 adults. The location is the problem: people who show up at a library in their free time are unusually likely to be heavy readers, so the sample mean probably sits above the true citywide average, making the estimate biased upward and unrepresentative. The claim that a mean is always the true population value is false; a sample mean is an estimate that shifts from sample to sample. The claim that 200 responses are enough regardless of how people were selected confuses precision with accuracy, since bias does not shrink as the sample grows. And surveying on a Saturday gives no equal chance to adults who never pass that library.
- In an observational study, people who spent more time outdoors reported higher happiness levels. What can be concluded from the study?
Answer: There is a correlation between outdoor time and happiness.
This question asks what can be concluded from an observational study finding people who spent more time outdoors reported higher happiness. The data shows an association between outdoor activity and self-reported happiness levels. Answer B correctly identifies this as a correlation without claiming causation. Option A inappropriately claims causation, option C makes an absolute statement about all happy people, and option D claims outdoor activity is the sole determinant of happiness. Observational studies can identify correlations but cannot establish that one variable causes changes in another without experimental manipulation and controls.
- A student analyzed 12 months of data from her household and found that months with higher electricity use (kWh) also had higher water use (gallons). She concludes, "Using more electricity makes us use more water." Which critique is most appropriate?
Answer: The conclusion is not justified because the relationship could be due to a third variable such as hot weather increasing both air-conditioning and water use.
This question critiques a causal conclusion from correlated household electricity and water use over 12 months. The data shows a positive association between the two variables. Choice A is most appropriate because it highlights potential confounders like hot weather driving both, so correlation does not prove causation. Choice B oversteps by stating correlations always indicate causation, which is false. Choice C incorrectly assumes sufficient data eliminates confounders without evidence. For correlation critiques, stress that associations suggest but do not prove causality; always consider third variables.
