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.”  

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.

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.

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.

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.

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.

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.

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.

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).

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.