SAT Mathematics : Drawing Logical Conclusions

Study concepts, example questions & explanations for SAT Mathematics

varsity tutors app store varsity tutors android store varsity tutors amazon store varsity tutors ibooks store

Example Questions

Example Question #1 : Data & Conclusions

A student wants to find out whether mothers in his city support a local initiative to redesign the city’s largest park. He decides to conduct a poll on this question by calling the home phone numbers of all 300 students in his private high school, asking to speak to their mothers, and recording their responses as yes, no, or no preference. Of the calls, 250 were answered and 200 mothers responded “yes,” with 40 saying “no” and 10 saying “no preference.”

Does this survey lead to the conclusion that most mothers in the city support the initiative to redesign the park?

 

Possible Answers:

Yes, because there are five times as many “yes” answers as “no” answers.

No, because we do not know how the mothers who did not answer the phone would have responded.

No, because the mothers of private high school students may not be representative of all mothers in the city.

Yes, because 80% of the mothers who answered are in favor of the initiative.

Correct answer:

No, because the mothers of private high school students may not be representative of all mothers in the city.

Explanation:

Answer: No, because the mothers of private high school students may not be representative of all mothers in the city.

It is important to note here that the group of people the conclusion is about - “mothers in the city” - is not the same as the group of people surveyed. The people surveyed were mothers of private school students (which means we don’t know what mothers of public school students would have thought) and of high school students (so we don’t know what mothers of younger students would have thought). This creates too narrow of a sample to draw a conclusion about the larger population of “all mothers in the city.”  

 

Example Question #2 : Data & Conclusions

The marketing department of a chain of pet stores surveyed a random sample of American families about the number of pets they have. Using the sample data, the department estimates that 81% of families have at least one pet in their home. The margin of error for this estimation is 5%. Which of the following conclusions about all American families can most appropriately be drawn from this data?

Possible Answers:

The marketing department is between 76% and 86% confident that most American families have at least one pet in their home.

 

More than 76% of American families have exactly one pet in their home.

It is likely that between 76% and 86% of American families have at least one pet in their home.

No greater than 86% of American families have a pet in their home.

Correct answer:

It is likely that between 76% and 86% of American families have at least one pet in their home.

Explanation:

Answer: It is likely that between 76% and 86% of American families that have at least one pet in their home.

A margin of error creates a range between which the statistic is most likely to fall. Here the researchers’ best estimate of the percentage of American families with at least one pet is 81%, and if you add and subtract the 5% margin of error you arrive at a range of 76-86%, meaning that that range is where the true, exact percentage most likely falls. Note, however, that while it is very likely that the true statistic falls in that range it is not 100% guaranteed, so the conclusions that “no greater than 86%” or “more than 76%” go too far to be proven from this one survey.



Example Question #3 : Data & Conclusions

cafeteria service company serves lunches at all 20 schools in a particular school district. In its effort to satisfy students, the company seeks to learn how happy students are with the variety of cafeteria lunch offerings they have, but it suspects that happiness may vary school to school. Which of the following survey methods is most appropriate to estimate the percentage of students in the district who are happy with their cafeteria options?

Possible Answers:

Select 20 students from each school and survey each student selected about how happy they are with the variety of cafeteria options.

Select one of the schools at random and survey each student at that school about how happy they are with the variety of cafeteria options.

Visit each of the 20 schools for one lunch period each to observe how happy students are with their lunches.

Send out a survey to all students in the school asking about how happy they are with the variety of cafeteria options, and then use the first 20 responses that come in.

Correct answer:

Select 20 students from each school and survey each student selected about how happy they are with the variety of cafeteria options.

Explanation:

Answer: Select 20 students from each school and survey each student selected about how happy they are with the variety of cafeteria options. Particularly because the company believes that the happiness will range from school to school, it is important to have a sample from each school so that the company has insight into each school’s preferences. Note that while visiting each of the 20 schools might also give insight into the preferences at each school, observing that only one one day gives too narrow a sample to understand how students feel about the variety of options, since presumably each student will only be eating one meal that day.

Example Question #4 : Data & Conclusions

Central High School has gotten large enough that the district plans to create a second high school. When the district sent ballots to all of its students and teachers to choose a mascot for the new school, 20% of students and 80% of teachers voted. Of those who voted, 30% of the students and 70% of the teachers chose Sharks as the mascot, and 50% of students and 20% of teachers chose Bears as the mascot. Which of the following conclusions can properly be drawn from the data above?

 

Possible Answers:

If all students had voted, Bears would have gotten the greatest number of total votes.

More teachers than students participated in the voting for the new mascot.

More than half of all teachers chose Sharks as the new school’s mascot.

Sharks was the mascot that received the greatest number of total votes.

Correct answer:

More than half of all teachers chose Sharks as the new school’s mascot.

Explanation:

Answer: More than half of all teachers chose Sharks as the new school’s mascot.  We are told that 80% of all teachers voted, and of those 80% who voted 70% chose Sharks. That means that 56% of all teachers chose Sharks, which is more than half.

Of the other choices, note that we don’t know whether there are more students or teachers, and it’s at least possible that there are quite a few more teachers than students. If, for example, there are 1,000 students and 50 teachers, then that would mean 200 students voted and 100 of them chose Bears, whereas only 28 teachers chose Sharks -- the big advantage that Sharks has among teacher voters is inconsequential if the students far outnumber the teachers, so “Sharks received the greatest number of votes” and “more teachers than students participated” are not necessarily true. 

We also cannot conclude that if all the students had voted, Bears would have won: for one, we don’t know that students far outnumber the teachers, and secondly we also don’t know how those 80% of students who did not vote would have voted.

Example Question #5 : Data & Conclusions

To determine the average number of pets per household in the town of Los Gatos, a researcher surveyed 100 patrons of the town’s most popular pet supply store. Of the 100 people surveyed, the average number of pets per household was 1.8. Which of the following statements must be true?

Possible Answers:

It can logically be concluded that most households in Los Gatos have at least one pet.

The sampling method used by the researcher is flawed and may produce a biased estimate of the average number of pets per household in Los Gatos.

The researcher chose too small a sample size to make a reasonable conclusion about the number of pets per household in Los Gatos.

The average number of pets per household in Los Gatos is 1.8.

Correct answer:

The sampling method used by the researcher is flawed and may produce a biased estimate of the average number of pets per household in Los Gatos.

Explanation:

Answer: The sampling method used by the researcher is flawed and may produce a biased estimate of the average number of pets per household in Los Gatos.

An important consideration here is that the researcher only surveyed people who were already at a pet supply store, so the sample she chose consisted entirely of people very likely to have at least one pet. People without pets rarely ever go to buy pet supplied! So her sample is likely to give a biased estimate, since it likely left out lots of people who would have had 0 pets.

 

Example Question #6 : Data & Conclusions

Of the 80 ninth grade students Johanna surveyed at random at her high school, 27.5% of them stated that they prefer year-round school to their current schooling system. If this survey is representative of the 280 students in the ninth grade class, which of the following is closest to the number of students in the class who prefer year-round school?

 

Possible Answers:

80

70

20

100

Correct answer:

80

Explanation:

Very occasionally the SAT will give you more information within a problem than you need in order to solve it. In this case, you don’t need the information about the number of people Johanna originally surveyed. While this information would be useful if you were trying to determine the validity of the survey, it isn’t necessary to find the number of students who prefer year-round school. You are given the percent of students who prefer year round school and the total number of students, so you simply need to be able to find the answer to the question “what is 27.5% of 280?”

To do that, you simply need to translate the numbers given into math. Remember that “percent” just means divided by 100, and that the word “of” means to multiply, so the expression becomes:

 

There are a total of 77 students who prefer year-round schooling in the 9th grade at Johanna’s school. Although this isn’t a potential answer, don’t panic! The question asks which answer is closet to the number of students who support year round schooling. Since 77 rounded to the tens place is equal to 80, you can safely choose 80 as your answer.

Example Question #1 : Drawing Logical Conclusions

To determine how citizens feel about the city’s proposed plan to build a second library, researchers surveyed 50 library visitors and found that 42 of them supported the plan and 30 of them would even be open to a tax increase to fund it. Which of the following statements must be true?

Possible Answers:

The researcher’s sampling method was flawed and may produce a biased estimate of the popularity of the proposal.

The majority of the city’s citizens support the construction of a second library.

Because the researcher did not choose a large enough sample size, the results of the survey are questionable.

It is clear that most citizens are in favor of a second library, but unlikely that most citizens would support a tax increase to pay for it.

Correct answer:

The researcher’s sampling method was flawed and may produce a biased estimate of the popularity of the proposal.

Explanation:

Answer: The researcher’s sampling method was flawed and may produce a biased estimate of the popularity of the proposal. 

Whenever you’re looking at a survey, it is critical to ask whether the group surveyed is representative of the overall population that the survey is meant to learn about. Here the survey is intended to learn about “citizens” but the group surveyed is a narrow subset of that group: “library visitors.”  And, of course, who is most likely to be in favor of more libraries? Presumably the people that actively use the library. So it is very likely that this survey would overestimate the proportion of overall citizens who favor a new library - it’s only surveying library users and does not include the results of anyone who does not.

Among the other answer choices, note that depending on the size and composition of the city, a sample size of 50 may be appropriate to draw conclusions so it is not the size of the sample that’s the issue, it’s the fact that it’s biased toward one subset of the population. And because of that bias, we cannot draw either of the conclusions about support for the library.

 

 

Example Question #8 : Data & Conclusions

The mayor of a city wants to determine whether the city’s citizens would support a small tax increase to expand and renovate the city’s playgrounds. She randomly surveyed 200 parents who live in the city and found that nearly 75% of them would be in favor of the proposal. Which of the following is true of the survey?

 

Possible Answers:

The survey should have consisted exclusively of citizens who are not parents.

Because it did not include any citizens who are not parents, it is likely that the popularity of the proposal may be overestimated.

Because it only surveyed people who live in the city, whereas people who live outside the city limits might also visit the playgrounds, the survey’s methodology is flawed.

The survey indicates that if the proposal were to be put up for a vote of the city’s citizens, it would win a majority of votes.

Correct answer:

Because it did not include any citizens who are not parents, it is likely that the popularity of the proposal may be overestimated.

Explanation:

Answer: Because it did not include any citizens who are not parents, it is likely that the popularity of the proposal may be overestimated.

This survey methodology is guilty of a very common survey error that you will see on the SAT: the mayor wants to find out about a different group of people (the city’s citizens) than were surveyed (parents who live in the city). Parents are a subset of citizens, and in this case they are likely to be more in favor of playgrounds than non-parents, since typically children are the people who use playgrounds, and parents are likely to want their children to have better playgrounds to use.  Because the citizenry likely includes nonparents, this focus only on parents creates a potentially-biased sample.

Among the other answer choices, focusing exclusively on non-parents would create a similar issue of a non-representative sample. Because the sample is biased, we cannot conclude that the proposal would garner the majority of votes. And the idea that people outside the city might also use the playground doesn’t change the fact that the mayor wants to know what citizens think, so not considering those who live elsewhere doesn’t bias the sample from what the mayor really wants to know.

 

Example Question #9 : Data & Conclusions

For a school project, a student randomly selected hundreds of books from the public library to learn about their characteristics. Among the sample were 200 novels, only 20% of which were 300 pages or longer. Which of the following conclusions is best supported by the data?

Possible Answers:

Only 20% of the books at the public library are longer than 300 pages.

Less than half of the novels from the public library are longer than 300 pages.

Approximately 80% of all books are less than 300 pages in length.

Most books at the public library are less than 300 pages long.

Correct answer:

Less than half of the novels from the public library are longer than 300 pages.

Explanation:

Whenever you approach data problems that ask you to draw or analyze conclusions, it is extremely important to note the specific language of the sample that the data corresponds to. Here we see that the student is analyzing books from the public library, but the only books we're given data about is the 200 novels that were part of the sample, and novels are only one type of books. So we cannot draw conclusions about "all books" (which would include dictionaries, biographies, etc. in addition to novels). We do know, however, that since only 20% of novels are 300 pages or longer, then less than half (20% is less than 50%) of novels are longer than 300 pages. (And presumably some of those 20% might even be exactly 300 pages, meaning that "longer than 300 pages" could be even less than that 20% that includes 300 or longer).

Example Question #1 : Data & Conclusions

A marine biologist randomly tagged and measured sharks in the Atlantic Ocean to study their characteristics and habits. The sample included 150 hammerhead sharks, of which 60% measured more than 12 feet in length. Which of the following conclusions is best supported by the data?

Possible Answers:

Less than half of the hammerhead sharks in the Atlantic Ocean are longer than 12 feet long.

More than half of the sharks in the Atlantic Ocean are longer than 12 feet long.

Hammerhead sharks are among the longest sharks in the Atlantic Ocean.

Approximately 40% of the shark population in the Atlantic Ocean is 12 feet long or less.

Correct answer:

Less than half of the hammerhead sharks in the Atlantic Ocean are longer than 12 feet long.

Explanation:

An important consideration in this problem is that, while the marine biologist researched "sharks" in general, we only have measurement data for 150 hammerhead sharks, a particular type of shark. So we cannot draw any statistical conclusions about sharks in general, since we only know about one type of them. 

We do know that 60% of the hammerheads studied are more than 12 feet long, meaning that more than half measure longer than 12 feet. That also means that the other portion, less than half, do not measure longer than 12 feet. And since we have a random sample of hammerheads, we can extrapolate this conclusion to all hammerheads, so we can conclude that less than half of the hammerheads in the Atlantic are longer than 12 feet.

Learning Tools by Varsity Tutors