SAT Math Quiz: Inferences And Claims From Statistics

What this quiz covers

This quiz focuses on Inferences And Claims From Statistics, giving you a quick way to practice the rules, question types, and explanations that matter most for SAT Math.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

Question 1

1. The claim may be unreliable because the responses were voluntary, so nonresponse bias could make the 240 respondents unrepresentative of all 3,000 students. (correct answer)
2. The claim is definitely true because 240 is a large enough sample size to eliminate bias, even when participants choose whether to respond.
3. The claim is definitely false because a survey can never be used to estimate a population proportion without interviewing every student.
4. The claim is justified because emailing all students guarantees a random sample, regardless of how many students respond.

Explanation: This question addresses nonresponse bias in survey sampling. While the researcher emailed all 3,000 students (attempting a census), only 240 responded voluntarily - an 8% response rate. The key issue is that students who chose to respond might differ systematically from non-respondents; perhaps satisfied students were more likely to respond, or those with strong opinions either way. This voluntary response creates nonresponse bias, making the 240 respondents potentially unrepresentative of all students. The 68% satisfaction rate among respondents may not reflect the true population proportion. Choice A correctly identifies nonresponse bias as the critical limitation, while choice B incorrectly assumes sample size alone eliminates bias from voluntary response.

Question 2

1. The analysis suggests heavier viewing is associated with higher bills in this region, but plan type or pricing rules could explain the difference without viewing causing it. (correct answer)
2. Because the sample size is large, the analysis proves that increasing viewing time will raise any customer's bill, regardless of plan or region.
3. Because the accounts were not randomly sampled, the analysis shows there is no relationship between viewing time and monthly bill for any customers.
4. The analysis proves that customers with low viewing time are overpaying, since their bills are lower on average than heavy viewers' bills.

Explanation: This question examines whether viewing time causes higher bills based on observational data from a streaming service. The data show that accounts with more viewing time have higher average bills, but this is observational data without random assignment of viewing time. Critically, the analysis notes that prices vary by plan type in this region, which could explain the association - heavy viewers might choose more expensive plans with more features. The correct answer recognizes that while there's an association, other factors like plan type or pricing rules could explain the difference without viewing time causing higher bills. Choice B incorrectly claims causation from observational data, ignoring the confounding factor of plan types.

Question 3

1. The data show a negative association in this volunteer sample, but they do not establish that increasing exercise will cause a lower resting heart rate for an individual. (correct answer)
2. The data prove exercise causes lower resting heart rate, because the scatterplot shows a clear downward pattern with no points above 80 bpm.
3. The data show that resting heart rate and exercise minutes are unrelated, because the points do not form a perfect straight line.
4. The results can be generalized to all adults in the city because 60 participants is enough to eliminate selection bias from volunteering.

Explanation: This question examines whether observational data about exercise and heart rate can support a causal claim. The scatterplot shows a negative association - people who report more exercise tend to have lower resting heart rates in this volunteer sample. However, because participants self-selected into the study and were not randomly assigned to exercise levels, we cannot conclude that increasing exercise will cause lower heart rates for any individual. The association could be explained by other factors like genetics, diet, or health consciousness that affect both exercise habits and heart rate. Choice B incorrectly claims causation from observational data, while choice A correctly distinguishes between observing an association and proving causation.

Question 4

1. Fertilizer X will increase the height of all plant species in any environment because the experiment used plants and measured growth.
2. Fertilizer X caused plants to grow taller on average than no fertilizer in this experiment, since treatment was randomly assigned and conditions were controlled. (correct answer)
3. All plants given Fertilizer X will be at least 4 cm taller than plants given no fertilizer because the group averages differ by 4 cm.
4. Fertilizer X is associated with taller plants, but the study cannot determine whether Fertilizer X caused the difference in height.

Explanation: The question asks what conclusion a randomized, controlled experiment supports. Because the 60 identical plants were randomly assigned to Fertilizer X or no fertilizer, and light and water were held constant, the only systematic difference between the groups is the fertilizer, so a difference in average final height can reasonably be attributed to the fertilizer itself. That is exactly what the statement crediting Fertilizer X with causing taller average growth in this experiment says. The statement that the study can show only an association and cannot determine cause understates what random assignment buys you; that caution belongs to observational studies. The claim covering all plant species in any environment generalizes far beyond the single species and single set of conditions tested. And the claim that every treated plant will be at least 4 cm taller misapplies a difference in group averages to individual plants, which averages never guarantee.

Question 5

1. The survey proves headphones cause longer studying, because the mean study time is higher for owners than for non-owners.
2. The survey shows headphone ownership and study time are unrelated, because both groups studied for multiple hours.
3. The survey supports an association in this dorm, but it cannot show headphone ownership causes longer studying, because students were not randomly assigned to own headphones. (correct answer)
4. The survey proves all college students study 9.4 hours per weekend, because the sample includes 90 students.

Explanation: The survey compared self-reported study hours for headphone owners, averaging 9.4, against non-owners, averaging 7.8, among 90 students in one dorm. The honest evaluation is the one saying this supports an association in that dorm but cannot show ownership causes longer studying, because no one randomly assigned who owns headphones. Students who buy them may already be more motivated studiers, which explains the 1.6-hour gap without headphones causing anything. The statement that the survey proves headphones cause longer studying draws a causal conclusion that observational data cannot support. The statement that ownership and study time are unrelated because both groups studied several hours ignores the actual difference between the means. The statement that all college students study 9.4 hours generalizes one dorm's owner average to an entire national population from a non-random sample.

Question 6

1. All alert individuals drink coffee daily.
2. There is an association between coffee drinking and alertness. (correct answer)
3. Daily coffee consumption increases alertness.
4. Coffee is the only factor affecting alertness.

Explanation: The question asks what a survey of 1000 adults, in which daily coffee drinkers report higher alertness, actually justifies. A survey records what people already do and report; nobody was assigned to drink coffee and no other differences were controlled. So the defensible conclusion is the modest one: coffee drinking and alertness are related, which is an association. The statement that daily coffee consumption increases alertness makes a causal claim the design cannot support, since naturally alert people may simply choose coffee, or sleep, age, and work schedule could drive both. The statement that coffee is the only factor affecting alertness is stronger still, ruling out every other influence on no evidence. And the statement that all alert individuals drink coffee daily is an absolute about an entire population that a group-level tendency in one sample cannot establish.

Question 7

1. Students who read frequently tend to have better vocabulary scores. (correct answer)
2. All students who read more than 10 books will have high vocabulary scores.
3. Reading is the only factor influencing vocabulary scores.
4. Reading more than 10 books annually improves vocabulary scores for all students.

Explanation: The question asks which claim the data support: among 200 high school students, those reading more than 10 books a year scored higher on vocabulary tests. Higher average scores for one group describe a tendency within this sample, so the claim that students who read frequently tend to have better vocabulary scores is properly hedged and matches the evidence. The claim that all students who read more than 10 books will have high scores turns a group tendency into a guarantee about every individual, which averages never provide. The claim that reading improves vocabulary for all students asserts both causation and universality from an observational study; strong readers may already have larger vocabularies, or an engaged home may drive both. And the claim that reading is the only factor influencing vocabulary rules out schooling, conversation, and everything else with no supporting evidence.

Question 8

1. Eliminating coffee will reduce stress for all individuals.
2. There is no relationship between coffee consumption and stress.
3. People who drink more coffee tend to report higher stress levels. (correct answer)
4. Drinking coffee increases stress levels.

Explanation: 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.

Question 9

1. 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. (correct answer)
2. The sample must be unbiased because 500 is a large number, so it will always represent the state population well.
3. Generalization is guaranteed because phone surveys always reach a random sample of all voters in the state.
4. The survey will prove whether the transit tax causes people to vote more often because it asks about support for the tax.

Explanation: 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.

Question 10

1. The claim is justified because a positive correlation means documentaries cause longer sessions, and a sample of 3,000 eliminates alternative explanations.
2. The claim is false because $r=0.48$ is not exactly 1, and only a perfect correlation can indicate any relationship between two variables.
3. The data prove that longer sessions cause people to watch more documentaries, because correlation identifies which variable is the cause and which is the effect.
4. The data show an association between documentary watching and session length, but other factors (such as user preferences) could explain it, so causation is not established. (correct answer)

Explanation: The question asks how to evaluate the claim that watching documentaries makes people stay longer, given r = 0.48 across 3,000 accounts. A moderate positive correlation shows the two behaviors travel together, but nothing rules out a third explanation: viewers who enjoy documentaries may simply be the kind of users who settle in for long sessions. So an association is established and causation is not. The response accepting the claim because 3,000 accounts eliminate alternatives is wrong, since a larger sample improves precision but never converts a correlation into a cause. The response calling the claim false because 0.48 is not exactly 1 misreads the statistic; values between 0 and 1 indicate real but imperfect relationships. And the response asserting that longer sessions cause more documentary watching merely reverses the arrow, which correlation cannot settle.

Question 11

1. Because of random assignment, the study provides evidence that instrumental music can cause higher average reading speed for students similar to those in the class. (correct answer)
2. Since the music group read faster on average, the students in that group must have had higher baseline reading ability than the silence group.
3. Because the students were not randomly selected from the school, the study shows no relationship at all between music and reading speed.
4. The results prove that instrumental music will increase reading speed for all students in the district by 13 words per minute.

Explanation: Students were randomly assigned to read with music or in silence, and the music group averaged 245 words per minute against 232, a 13-word difference. Random assignment is what licenses a causal reading, so the justified statement is the one saying the study gives evidence that instrumental music can cause higher average reading speed for students similar to those in the class. The statement that the music group must have had higher baseline ability works against the design, since randomization is precisely what makes systematic baseline differences unlikely. The statement that the study shows no relationship at all because students were not randomly selected confuses two ideas: non-random selection limits how widely the result generalizes, but it does not erase the measured difference. The statement that the results prove a 13-word gain for all district students overstates both certainty and reach.

Question 12

1. Because the question concerns voters, the sample should include only people who voted in the last election, so the 54% result is invalid for registered voters.
2. The sample may be biased because registered voters without listed landlines or who are unavailable on weekday afternoons could be underrepresented in the estimate. (correct answer)
3. Because 1,200 is a large sample, the sampling method cannot introduce bias, so 54% must be the true support level for all voters.
4. Because the researcher is estimating a proportion, she must measure each voter's exact level of support on a 0–100 scale, not a yes/no response.

Explanation: Generalizing from the people reached to all registered voters requires a sample that can reach the whole population, and calling listed landline numbers on weekday afternoons cannot. The correct choice names that coverage gap: voters with no listed landline, including cell-only households, and voters working during weekday afternoons have little chance of being sampled, so 54% may not describe all registered voters. The choice arguing that 1,200 is large enough to rule out bias confuses precision with accuracy, since a big sample drawn from a skewed frame is just a precisely wrong estimate. The choice demanding an exact 0-100 support scale is unnecessary, because a proportion is estimated from yes-or-no responses by definition. The choice restricting the sample to people who voted last election changes the target population from registered voters to past voters.

Question 13

1. The study used two groups, so it cannot show any relationship between app use and quiz scores.
2. The average quiz score is reported, so it is impossible to compare the two groups' performance.
3. The study lasted 3 weeks, so the results automatically generalize to all future school years.
4. Students self-selected into groups, so motivation or prior achievement could differ between groups and explain the score difference. (correct answer)

Explanation: The question asks what most weakens the conclusion that the app caused the 88-versus-82 quiz gap. Students chose their own groups, so app users and non-users may have differed before the study began, in motivation, prior achievement, or study habits, and any of those differences could produce a six-point gap by itself. That self-selection problem is why random assignment is what licenses a causal claim. The statement that using two groups makes it impossible to show any relationship is wrong; comparing groups is precisely how relationships get detected. The claim that reporting an average makes comparison impossible is also wrong, since a mean is a standard way to compare group performance. And the statement that a three-week study automatically generalizes to all future school years asserts more than the data support rather than identifying a limitation.

Question 14

1. The scatterplot shows that exactly 10 more minutes of homework increases quiz averages by exactly 5 points for all students, because the pattern is roughly linear.
2. The scatterplot proves there is no relationship between homework and quiz scores, because the points do not form a perfect straight line.
3. Increasing homework minutes will cause every student in the school to raise their quiz score, because the plotted points generally rise from left to right.
4. In this honors class, higher reported homework time is associated with higher quiz averages, but the plot alone cannot show that homework time causes higher scores. (correct answer)

Explanation: The question asks what a plot of self-reported homework minutes against quiz averages, for 28 students in one honors class, can justify. An upward trend supports an association: within this class, higher reported homework time goes with higher quiz averages. It cannot support causation, because the teacher assigned no homework amounts, so students chose their own study time and factors like prior preparation, tutoring, or motivation could drive both variables. The statement that exactly 10 more minutes raises quiz averages by exactly 5 points for all students invents a precise universal rate that a loose trend with widely spread points cannot deliver. The claim that more homework will cause every student in the school to raise scores both assumes causation and generalizes beyond one honors class. And the claim of no relationship misreads imperfect linearity, since points need not form a straight line to show a trend.

Question 15

1. Among these members, app use was associated with more gym visits on average, but the difference could be explained by pre-existing motivation. (correct answer)
2. Most people will visit the gym 2.3 more times per month if they download the app, regardless of their exercise habits.
3. The results show that gym visits and app use are unrelated, since the groups had different sample sizes.
4. Using the app causes members to visit the gym more often, because the app-user average visits were higher than the non-user average visits.

Explanation: App users averaged 9.1 visits and non-users averaged 6.8, a difference of 2.3 visits, but members chose for themselves whether to use the app. That self-selection is why the justified claim is the one reporting that app use was associated with more visits among these members while noting that pre-existing motivation could explain the gap. The claim that using the app causes more visits asserts causation from an observational design with no random assignment. The claim that most people will visit 2.3 more times if they download the app, regardless of habits, turns a group-average difference into a guaranteed individual effect and generalizes past the 90 members studied. The claim that visits and app use are unrelated because the group sizes differ misuses sample size: unequal group counts of 35 and 55 do not erase a difference in averages.

Question 16

1. The sign caused higher sales, because most stores had higher sales after the sign was added.
2. The sign had no relationship to sales, because 2 stores did not increase and therefore the sign cannot matter.
3. Sales tended to be higher after the sign was added, but seasonality and manager timing are confounders that limit causal conclusions. (correct answer)
4. The sign will increase sales at every store in the country, since 6 out of 8 stores increased in this comparison.

Explanation: Sales rose in 6 of the 8 stores after the sign went up, so something did trend upward, but managers chose when to add the sign and the change coincided with the start of the holiday season. Both are confounders: holiday traffic alone could lift sales, and managers may have added signs when they already expected a strong stretch. That makes the statement that sales tended to be higher afterward, while naming seasonality and manager timing as limits on causal conclusions, the accurate one. The statement that the sign caused higher sales ignores those alternative explanations. The statement that the sign had no relationship because 2 stores did not increase demands perfect consistency, but a tendency never requires every store to improve. The statement that the sign will increase sales at every store in the country generalizes from 8 non-random stores.

Question 17

1. Participating in sports decreases GPA.
2. Athletes who practice more may have slightly lower GPAs. (correct answer)
3. All students involved in sports will have poor academic performance.
4. Athletic practice has no effect on academic performance.

Explanation: This question asks which claim is supported by data showing a weak negative correlation between hours of athletic practice and GPA among 300 high school athletes. The study found that students with more practice hours tend to have slightly lower GPAs, indicating a modest inverse relationship. Choice B appropriately describes this correlation using cautious language ("may have slightly lower") that reflects both the negative relationship and its weak strength.

Question 18

1. Asking a different question in the survey.
2. Selecting a sample of voters only from one specific region.
3. Increasing the sample size to 2,000. (correct answer)
4. Decreasing the sample size to 500.

Explanation: The question asks which design change would most effectively shrink a 3% margin of error. Margin of error scales with 1/sqrt(n), so it shrinks only when the sample grows: going from 1,000 voters to 2,000 divides the margin by about sqrt(2), bringing it to roughly 2.1%. Cutting the sample to 500 does the opposite, multiplying the margin by about sqrt(2) to roughly 4.2%, so it makes the estimate less precise. Restricting the sample to one specific region does not reduce error about all state voters; it introduces selection bias, producing an estimate that is precise about the wrong population. Asking a different survey question changes what is being measured rather than the precision of the 52% figure, because precision depends on sample size, not wording.

Question 19

1. The study establishes causation between sleep quality and memory.
2. The study only shows a correlation, not causation, between variables. (correct answer)
3. The results apply to all age groups, not just college students.
4. Memory retention does not correlate with sleep quality.

Explanation: This question asks which limitation affects the conclusion of a study showing a positive correlation between sleep quality and memory test scores among college students. The fundamental limitation of correlational research is that it cannot establish causation, only association between variables. Choice B correctly identifies this core limitation, noting that the study demonstrates correlation but not causation between sleep quality and memory performance. Choice A incorrectly suggests the study establishes causation, choice C addresses generalizability rather than the causation limitation, and choice D contradicts the stated positive correlation findings. Understanding the difference between correlation and causation is essential when evaluating the strength and limitations of research conclusions.

Question 20

1. All people should get a pet to lower depression.
2. Owning a pet reduces depression levels.
3. Pet owners are less likely to be depressed. (correct answer)
4. Depression can be cured by owning a pet.

Explanation: In a sample of 500 people, pet owners reported lower average depression scores. Nobody was randomly assigned to own a pet, so the study can establish an association only, which makes the statement that pet owners are less likely to be depressed the strongest supportable conclusion: it describes the observed pattern without explaining what produced it. The statement that owning a pet reduces depression asserts cause and effect, which observational data cannot show; people who are less depressed may be more able to take on a pet, or income and living situation may influence both. The statement that depression can be cured by owning a pet goes further still, promising a cure the study never measured. The recommendation that all people should get a pet converts a group average into universal advice for individuals.