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.

This probability problem involves the concept of expected value. If the bakery is four times as likely to sell a chocolate chip cookie as it is to sell a snickerdoodle, then you can say that chocolate chip cookies are ⅘ of the total number of cookies sold, and that snickerdoodles are ⅕ of the total cookies sold.

This means that ⅘ of the cookies carry a 10% probability of a bonus brownie, and ⅕ of the cookies carry a 25% probability. You can then turn this into an equation:

⅘ (10%) + ⅕ (25%) is the expected probability of a random cookie winning the brownie. That means that the answer is 8% + 5% = 13%.

Whenever you're dealing with "pairs probability" it is important to ask yourself whether your job is to calculate the probability of a specific pair (e.g. a pair of red shoes) or of just "any" pair whatsoever. Here the question asks for "a pair" and not a specific pair, so it is important to recognize that the first shoe the dog pulls out can be any shoe, but the second shoe he pulls must match the first. So the only pull that "matters" is the second one - on that draw he must match the one he drew first.

At that point there are 23 shoes left (everything but the one he picked first), and only one that will match the first one (since we're dealing with 12 different pairs). This means that the probability is .

The probability of selecting a green marble at random is, conceptually, the number of green marbles divided by the number of total marbles. Since 2 green marbles were added to the original total of  green marbles, the number of green marbles to select is . And the total number of marbles is calculated as the starting total  plus the 2 green and 14 red marbles added, making the new total . Therefore, the correct answer is the new number of greens plus the new number total: .

This problem involves two critical elements of probability: dependent probability (when the outcome of one event then impacts the probability of the next event, because there are now fewer items to choose from) and sequential probability (when you're asked for the probability of multiple events happening in sequence, like "chocolate, then chocolate" here).

To start, note that the probability of drawing a chocolate candy on the first draw is calculated by taking the 6 chocolate candies and dividing by the 10 total candies, so .

For the second draw, there will only be 9 candies left (he will have already drawn one), and in the  of outcomes in which he has already chosen chocolate, there will be 5 chocolate candies left. So that probability is .

When you need a particular sequence of events to happen (chocolate, then chocolate) you multiply those probabilities together to find the probability of that sequence. That would mean that the probability is then:

The answer, therefore, is .

To find the chances of picking a guard, our formula should be the number of guards divided by the total number of players. We start with  guards and  forwards, and then add 2 guards and subtract one forward. So after the roster changes, we have  guards, and we have  total players. Since the probability of an outcome is the number of options that give you that outcome (guards) divided by the number of total options (all players), our probability is then .