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Inference from samples and studies

Judge what a sample can support, including margin of error and when a study can claim cause.

Problem-Solving and Data AnalysisMath
Practice inference from samples and studies free

What the SAT actually asks

You reason about study design rather than compute. Questions give a survey or experiment and ask which conclusion is supported. That depends on how the sample was chosen: a random sample from a population lets you generalize to that population and no further, and random assignment to treatment groups is what allows a claim about cause. Other questions give a margin of error and ask for the plausible range of the true value, or ask what would make that interval narrower.

Inference questions are relatively rare, usually about 1 of the 44 math questions, and some tests may not include one.

Where students lose points

How to attack these questions

  1. 1Identify the population the sample was actually drawn from, and refuse any conclusion that reaches beyond it.
  2. 2Check for random assignment before accepting any answer choice that uses causal language.
  3. 3For margin of error, build the interval by subtracting and adding the margin to the reported estimate.

Example questions

Written for this skill, with the answer explained. 49 more are in the app.

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Inference from samples and studies
Real question
020406080MapleOakPineBirchSpeciesTrees in the sample

A park has about 5,000 trees. A researcher identified the species of 200 of them, selected at random, with the results shown in the bar graph. Based on this sample, about how many of the park's 5,000 trees are oak?

Medium

A quality inspector selected a random sample of bolts from a production run and computed a 95 percent confidence interval of 7.92 millimeters to 8.08 millimeters for the mean diameter of bolts in the run. Which conclusion is most appropriate?

  • A) Every bolt in the run has a diameter between 7.92 and 8.08 millimeters.
  • B) Ninety-five percent of the bolts in the run have a diameter of exactly 8.00 millimeters.
  • C) It is plausible that the mean diameter of all bolts in the run is 8.00 millimeters. ✓
  • D) The mean diameter of the sampled bolts is unknown but probably near 8.00 millimeters.

Why: The interval gives the plausible values for the mean diameter of all bolts in the run, from 7.92 to 8.08 millimeters. Since 8.00 lies inside that range, a population mean of 8.00 millimeters is plausible. Saying "Every bolt in the run has a diameter between 7.92 and 8.08 millimeters." reads an interval about the mean as a range for every individual bolt, and single bolts vary more than the mean does. Saying "Ninety-five percent of the bolts in the run have a diameter of exactly 8.00 millimeters." turns the confidence level into a share of bolts at one exact diameter, which the interval says nothing about. Saying "The mean diameter of the sampled bolts is unknown but probably near 8.00 millimeters." calls the sample mean unknown, and the sample mean is known exactly; it is the center of the interval, 8.00 millimeters.

Medium

An agricultural scientist randomly sampled fields in one province and computed a 95 percent confidence interval of 3.8 to 4.6 metric tons per hectare for the mean wheat yield in that province. Which of the following would be an inappropriate use of this interval?

  • A) Reporting 4.2 metric tons per hectare as the sample mean for the province
  • B) Concluding that the mean wheat yield in a neighboring province is between 3.8 and 4.6 metric tons per hectare ✓
  • C) Concluding that a mean yield of 4.0 metric tons per hectare is plausible for the province
  • D) Concluding that a mean yield of 5.0 metric tons per hectare is not plausible for the province

Why: The interval estimates the mean wheat yield for the province that was sampled. Applying 3.8 to 4.6 metric tons per hectare to a neighboring province uses it for a population no field came from, which is the inappropriate use here. Saying "Reporting 4.2 metric tons per hectare as the sample mean for the province" reports the midpoint of 3.8 and 4.6 as the sample mean, which is exactly what the center of the interval is. Saying "Concluding that a mean yield of 4.0 metric tons per hectare is plausible for the province" calls 4.0 plausible, and 4.0 lies inside the interval, so that use is correct. Saying "Concluding that a mean yield of 5.0 metric tons per hectare is not plausible for the province" rules out 5.0, and 5.0 lies outside the interval, so that use is correct as well.

Harder

An analyst wants to cut the margin of error of a survey estimate roughly in half while keeping the same confidence level. Which change is most likely to accomplish this?

  • A) Increasing the sample size to about twice its current value
  • B) Keeping the sample size the same but repeating the survey twice
  • C) Increasing the sample size to about four times its current value ✓
  • D) Reducing the sample size to about half its current value

Why: The margin of error shrinks in proportion to 1 over the square root of the sample size. To cut it in half you need the square root of the sample to double, so the sample itself must be about four times as large. Saying "Increasing the sample size to about twice its current value" doubles the sample, which divides the margin by the square root of 2, about 1.4, not 2. Saying "Keeping the sample size the same but repeating the survey twice" runs the same-sized survey twice, which leaves each estimate's margin unchanged. Saying "Reducing the sample size to about half its current value" halves the sample, which widens the margin by about 1.4 instead of shrinking it.

Questions students ask

What does margin of error mean?

It gives the range around a sample estimate where the true population value plausibly falls. An estimate of 42 percent with a margin of error of 3 percent means the plausible range is 39 percent to 45 percent. It reflects sampling variability, not mistakes in the survey.

How do you make a margin of error smaller?

Increase the sample size. A larger random sample produces a narrower interval around the estimate. Changing the wording of a question or the confidence level is not what these SAT questions are testing.

When can a study conclude that one thing causes another?

Only when subjects were randomly assigned to the groups being compared, which is what makes it an experiment rather than an observational study. Without random assignment, the most you can say is that the two variables are associated.

Drill this skill until it is automatic

Questions on inference from samples and studies alone, with explanations on every one, and the app tracks which ones you keep missing.

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