Occasionally (less than one question per test on average), the SAT asks a question focused on how statistical studies work and what general conclusions can be drawn from them. These questions tend not to focus on specific statistical ideas, like percents, medians, or probability, but rather on the careful analysis of data used to draw appropriate general conclusions. The SAT uses these questions to test whether students can avoid overly general or specific statements in favor of the more cautious, qualified claims that can be appropriately drawn from studies. To this end, the concept of margin of error is sometimes introduced to reinforce the level of uncertainty that must be built into statistical methodology.
Although these questions are not frequent on the SAT, they are worth covering in a separate lesson because of 1) how rarely students encounter them in high school math programs outside of certain advanced statistics classes and 2) how different they feel from virtually any other kind of SAT math question.
A survey was conducted to determine how many of the 885 students at West Main High School like the school cafeteria food. One hundred students were selected for the survey, 25 for each of the four grades, and all responded. Of those 100, 47 indicated that they liked the school cafeteria food. Which of the following can be concluded based on this information?
I. If the entire school were surveyed, of students would affirm they like the school cafeteria food.
II. If another 100 students were selected with equal numbers of students from each grade, that survey would also use a figure of for students who like the school cafeteria food.A. Neither
B. I only
C. II only
D. Both I and II
Questions like this are a window into what stands behind the statistics one might see in an article or video about an election or other public issue. Have you ever wondered, How many people did they actually speak to in this survey? There is a whole field of statistics behind this practice that you don’t need to understand for the SAT. What you do need to understand is that a sufficiently large portion of a population can be used to talk about the population as a whole … as long as it is not claimed that the proportions identified in the smaller population are exactly transferable to the whole group.
Consider this question in particular. As long as there are more than 100 students total at West Main School (a fact assumed in the wording of the question), we cannot assume that of all students like the cafeteria food just because of those surveyed do. We are allowed to infer that approximately half of the whole student body likes the food; if given a margin of error, we can be even a little more specific (see questions below with margin of error). But otherwise, we must stay away from any statements that refer to exactly , unless we’re talking about the original 100 students surveyed.
This reasoning takes us all the way to our answer. We have already stated why Statement I cannot be inferred, but Statement II is no better; we know nothing exact about the next 100 students we might survey, even if they are also drawn evenly from all four grades. We can only make an educated guess that the “liking the food” portion would also be approximately half. The answer is Neither.
Final note: surveying only 100 students is a perfectly sufficient number to draw some conclusions about even the much larger total of 885 students. This is known as a representative sample. Unless the SAT presents a really tiny sample, like 5 people surveyed out of 5,000, you can assume that samples for surveys on the SAT are representative.
The main variation you’ll see in Methods and Inferences questions is a question that asks how far you can generalize certain results. You’ll get to practice this idea in one of the exercises below, but briefly, you can only generalize results to the original population targeted by the survey. For example, a survey of seniors (65 years old and older) in a certain community cannot tell you anything about the entire population of the community–much less about the population of the entire county, region, or state.
450 people work at a factory. A random sample of these workers was selected, and the selectees were asked whether they would accept a pay decrease in return for the establishment of a pension plan. of those selected said they would accept the pay decrease in return for the pension plan. Based on this survey, which of the following is the best estimate of the total number of workers at the factory who would accept the pay decrease in return for the pension plan?
A. 30
B. 45
C. 75
D. 135
The answer is 135. A question like this explores what conclusions can be drawn from a well-conducted survey. As long as the question says something like “random sample,” you can assume the survey is sound and that an inference can be drawn. Meanwhile, the fact that the question asks for an “estimate” means we are on solid ground, not in danger of drawing too exact an inference. In other words, we are safe to proceed with an actual calculation.
That calculation comes, as it often does in statistics, by means of a proportion. We know that if of the selected factory workers feel a certain way, roughly that same proportion of the whole population will feel that way. We do not have to use an actual algebraic proportion here because we are already given a percent: (percents are inherently expressions of proportion). We simply need to multiply that by the total population of . As discussed in the lesson on percents, we convert to and multiply that by , yielding our answer.
A firm conducted a survey of a seaside town to determine how many days per year the average resident spent at the beach. Based on the responses, the average was found to be 27 days, with a margin of error of 4 days. Which of the following is the best conclusion from these data?
A. of the town’s residents spent from 23 to 31 days per year at the beach.
B. Most of the town’s residents spent exactly 27 days per year at the beach.
C. It is not possible that any town resident spent less than 23 days at the beach.
D. It is plausible that most town residents spent between 23 and 31 days, inclusive, at the beach.
The answer is It is plausible that most town residents spent between 23 and 31 days, inclusive, at the beach.. The discussion found in the Introduction and the Approach question helps make sense of this question. Perhaps you have already internalized the core strategy found in this lesson: watch out for overly specific answers!. That principle alone eliminates all the wrong answer choices here, as each one has a marker of exactness: “”, “exactly”, and “not possible.” The best answer will use something like “plausible” or “likely” and provide a range of numerical possibilities that first subtracts the margin of error from the survey result () and then adds it (). That’s exactly what the right answer does here.
A middle school serves students in 6th, 7th, and 8th grades. A random sample of 7th-graders was selected and asked about their favorite local restaurant. What is the largest population to which the results of the survey can be generalized?
A. All 7th-graders at this middle school
B. All students at this middle school
C. All 7th-grade students nationwide
D. All middle school students nationwide
The answer is All 7th-grade students at this middle school.. The rule for this kind of question is that we can only extend the validity of survey results to the population initially concerned in the survey. The survey not only limited itself to students at a particular middle school, but it also narrowed its focus to 7th-graders alone. Remember, we can draw a conclusion about 7th-graders at this school even if not every student was surveyed, as long as the survey is randomized. We can extend our conclusions from this survey as far as 7th-graders at this school … but no further!
At a massive auction lot with thousands of cars, a study was done on the different weights of the cars. All of the 200 cars randomly selected were Hondas; weighed more than 4,000 pounds. Which of the following conclusions is best supported by the sample data?
A. Approximately of all the Hondas on the lot weighed more than 4,000 pounds.
B. Approximately of all the cars on the lot weighed more than 4,000 pounds.
C. Exactly of the Hondas on the lot weighed more than 4,000 pounds.
D. Less than half of all the cars on the lot weighed less than 4,000 pounds.
The answer is Approximately of all the Hondas on the lot weighed more than 4,000 pounds.. This question is exploring the same idea as the previous question, in a more specific way. Notice that the survey is done appropriately (randomly) but is narrow in scope: it focuses only on Hondas. Taking the principle we’ve already learned, we know that we can’t extend our conclusion beyond Hondas to all of the cars on the lot (nor can we conclude anything about all Hondas in the world generally).
With this in mind, we can narrow down our answers to the two choices that mention Hondas specifically. But we also know to avoid the word “exactly”, so the answer must be the one with the appropriately cautious “approximately” applied to Hondas specifically.
| “I go out to eat …” | Almost always prepares dinner at home | Regularly prepares dinner at home | Occasionally prepares dinner at home | Total |
| Rarely | 133 | 33 | 1 | 167 |
| Sometimes | 44 | 143 | 30 | 217 |
| Often | 7 | 35 | 31 | 73 |
A study asked 455 randomly selected single adults in the U.S. about their dinner habits–how often they prepare their dinner at home versus going out to eat. Based on the survey, an estimate is made that of U.S. single adults prepare their dinner at home regularly or often. The associated margin of error for this calculation is . Which of the following is a correct statement based on the given margin of error?
A. Approximately of those who replied to the survey were not being truthful.
B. It is impossible that the number of U.S. single adults who prepare their dinner at home at least regularly is .
C. The percentage of all U.S. single adults who only occasionally prepare dinner at home is .
D. It is doubtful that the number of U.S. single adults who only occasionally prepare dinner at home is .
The answer is It is doubtful that the number of U.S. single adults who only occasionally prepare dinner at home is .. A previous showed how margin of error works: we subtract the margin of error from the survey result to find the likely minimum result, and we add the margin of error to find the likely maximum possible. It’s worth stressing again: these results are only “likely”; we do not know for sure there will not be “outliers” below or above our range.
In this case, then, we take the from the survey and do our subtraction and addition to get a range of to . It is probable that the correct result for all U.S. single adults is in that range. Meanwhile, by subtracting these figures from 100%, we can infer that between and of U.S. single adults belong in the category or categories not included in the original (approximate) .
This reasoning puts us in a position to evaluate the choices. The answer choice mentioning regarding those who only occasionally prepare dinner at home seems good at first, but upon further reflection, it is too exact: there are no qualifying words like “approximately”. Meanwhile, although the answer with makes sense in its negative language, since is outside our desired range, we know not to select an answer that says “impossible”. Finally, the answer that talks about “ … not being truthful” misunderstands margin of effort entirely; the concept is never applied to the honesty or reliability of those being surveyed. That leaves the correct answer, which uses the appropriate word “doubtful” and focuses on the population that only occasionally makes dinner at home. That population, indeed, is unlikely to number more than of the total.
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