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Math: Problem-Solving and Data Analysis, 3 original questions

SAT Sample Inference and Margin of Error Practice: Questions and Answers

Interpret survey estimates, margin-of-error intervals and sample designs in three original SAT-style questions. Separate population inference from causation.

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The short answer

A sample statistic estimates a quantity in the population from which the sample was appropriately selected. A reported margin of error describes sampling uncertainty around that estimate under the study's stated assumptions; it does not fix biased selection or guarantee an exact population value. Add and subtract the margin to form an interval, keeping percentage points distinct from a percent change. Larger comparable random samples generally reduce sampling uncertainty. Random sampling supports population inference, while random assignment to treatments supports a causal comparison.

Method at a glance

  1. 1Identify the target population and design
  2. 2Build the reported interval
  3. 3Compare samples under stated conditions
  4. 4Separate generalization from causation

Try the 3 questionsRead the full method

Step-by-step method

Use these steps in order, then apply them to the questions below.

  1. Step 1: Identify the target population and design

    Determine who could be selected and whether selection was random. A large convenience sample can still miss the population you want to describe.

  2. Step 2: Build the reported interval

    Subtract and add the margin of error to the estimate using the same units. A margin of 4 percentage points around 62% gives 58% to 66%.

  3. Step 3: Compare samples under stated conditions

    Hold the confidence level and sampling assumptions comparable. A larger random sample generally gives a smaller margin; use a specific formula only when it is given or justified.

  4. Step 4: Separate generalization from causation

    Random sampling and random treatment assignment answer different questions. An observational association alone does not establish that changing one variable causes the other.

Try 3 SAT-style questions

These are original SAT-style questions written by PeakSAT, not official College Board questions. Choose an answer and check it to see why each choice is right or wrong, or open the worked answer.

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Question 1 of 3

A survey estimates that 62% of a town's adults favor a proposal, with a reported margin of error of 4 percentage points. Which interval corresponds to this estimate and margin?

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Correct answer

Choice B: 58% to 66%

Explanation

Subtract and add 4 percentage points to 62%: 62 − 4 = 58 and 62 + 4 = 66. The interval is 58% to 66%. This expresses the reported uncertainty; it is not a statement that the population percentage is known exactly.

Why each choice is right or wrong

  1. Choice A

    This uses a margin of 2 percentage points instead of 4.

  2. Choice B

    Correct answer

    Correct: the lower and upper endpoints are 58% and 66%.

  3. Choice C

    This includes only the lower side of the margin.

  4. Choice D

    This includes only the upper side of the margin.

Question 2 of 3

For two comparable random samples at the same confidence level, suppose the margin of error is proportional to 1/√n, where n is sample size. A sample of 400 has margin 6 percentage points. What margin does this model predict for a sample of 1,600?

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Correct answer

Choice B: 3 percentage points

Explanation

The sample size multiplies by 4, so √n multiplies by 2. Under the stated inverse-square-root model, the margin is divided by 2: 6/2 = 3 percentage points. This calculation relies on the comparability and model specified in the question.

Why each choice is right or wrong

  1. Choice A

    This divides the margin by 4 rather than by √4 = 2.

  2. Choice B

    Correct answer

    Correct: four times the sample size halves the margin under this stated model.

  3. Choice C

    The given model predicts a decrease when n increases.

  4. Choice D

    This reverses the relationship, increasing the margin when sample size grows.

Question 3 of 3

A well-conducted observational study randomly samples 200 adults from one town and records their exercise time and sleep duration. It finds a positive association. No exercise treatment is assigned. Which conclusion is most justified?

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Correct answer

Choice D: The study can inform the town's adult population, but the association alone does not establish causation

Explanation

Random sampling from the town supports cautious inference about that town's adult population. Because the study only observes existing behavior and does not assign treatments, other factors could explain the association. It does not establish that increasing exercise causes longer sleep.

Why each choice is right or wrong

  1. Choice A

    Observed association can be explained by other variables and is not itself proof of a cause.

  2. Choice B

    The sample was selected from one town, so worldwide generalization is not supported.

  3. Choice C

    The study did not test an intervention or determine each individual's response.

  4. Choice D

    Correct answer

    Correct: the sampling frame supports population inference, while the observational design limits causal claims.

Common mistakes

  • Calling a sample percentage an exact population percentage. Keep the estimate and its uncertainty visible.

  • Treating percentage points as a relative percentage of the estimate. Subtract the stated points directly when forming that interval.

  • Using random selection as proof of cause and effect. Causal comparison requires an appropriate study design, not just a representative sample.

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Common questions

Does a margin of error account for every survey problem?

No. It describes sampling uncertainty under assumptions. Nonresponse, biased selection or misleading questions can introduce other errors.

Does a larger sample always fix bias?

No. More observations from the wrong population can preserve selection bias even while a calculated sampling margin becomes smaller.

How are random sampling and random assignment different?

Random sampling selects who enters a study and supports population inference. Random assignment allocates treatments and helps support causal comparisons.

Sources

  1. College Board: Math section overviewsatsuite.collegeboard.org
  2. College Board: Student Question Bank Math skillssatsuite.collegeboard.org

Sources are linked for reference; they do not endorse PeakSAT. How we prepare our guides