How to solve SAT margin of error and sampling questions

This skill is about what a sample can and cannot tell you about a larger population. On the Digital SAT you use a sample proportion or mean to estimate a population value, interpret a margin of error, and judge which conclusions a sample supports. The math is light. The points come from reading the study description carefully. Problem-Solving and Data Analysis is about 15% of the Math section, or 5 to 7 of the 44 questions. Those questions are split across seven skills, so expect about one question on this skill, and occasionally none.

What the questions look like

Estimate a population total. A random sample shows that some fraction of people have a trait. You apply that fraction to the whole population to estimate how many people in it have the trait.

Interpret a margin of error. A study reports an estimate, such as a mean of 18.4 meters, with a margin of error. You choose the conclusion that the estimate and margin of error support.

Judge the sample. You decide which population the results can be applied to, whether a sampling method was biased, or how changing the sample size would affect the margin of error.

Step-by-step method

1. Identify the population that was sampled. Results from a random sample apply to the population the sample was drawn from, and only that population. A random sample of one city's voters says nothing reliable about voters in another city.
2. Check that the sample was random. Volunteers, people who answer an online poll, or customers who happen to walk by are not random samples. Results from these samples should not be generalized.
3. Scale up proportions. Sample proportion times population size gives an estimate for the population.
4. Turn the margin of error into an interval. The plausible range for the population value is the estimate minus the margin of error to the estimate plus the margin of error.
5. Interpret the interval as a statement about the population value. It describes where the population mean or proportion plausibly lies. It does not describe where individual values fall.
6. Know what changes the margin of error. With other conditions the same, a larger random sample gives a smaller margin of error, and a higher confidence level, such as 99% instead of 95%, gives a larger one.

Worked example

Example 1. A city has 52,000 registered voters. In a random sample of 400 of them, 140 said they support a new transit proposal. Based on the sample, about how many of the city's registered voters support the proposal?

The sample proportion is

$$\frac{140}{400} = 0.35$$

Apply it to the population:

$$0.35 \times 52{,}000 = 18{,}200$$

About 18,200 registered voters in the city support the proposal. This is an estimate for this city's registered voters only, because that is the population the sample came from.

Example 2. Researchers measured 150 trees selected at random from a large forest. The mean height of the trees in the sample was 18.4 meters, with an associated margin of error of 1.1 meters at the 95% confidence level. Which conclusion is best supported?

A) Every tree in the forest is between 17.3 and 19.5 meters tall.
B) The mean height of all trees in the forest is exactly 18.4 meters.
C) The mean height of all trees in the forest is plausibly between 17.3 and 19.5 meters.
D) The mean height of trees in every forest in the region is plausibly between 17.3 and 19.5 meters.

The interval is $18.4 - 1.1 = 17.3$ to $18.4 + 1.1 = 19.5$ meters. It estimates the mean height of the trees in this forest.

• A is wrong because the interval is about the mean, not about individual trees. Many trees are surely shorter or taller.
• B is wrong because a sample mean is an estimate, not an exact value.
• D is wrong because the sample came from one forest, so the results don't extend to other forests.

The answer is C.

Common traps

• Applying the interval to individuals. A margin of error describes uncertainty about a population mean or proportion. Choices that say "all" or "most" individuals fall in the interval are wrong.
• Overgeneralizing the population. If the sample came from one school, one city, or one forest, conclusions apply only there. Watch for choices that expand to "all students" or "the entire country."
• Trusting a self-selected sample. If people chose whether to respond, as in an online poll or a survey left at a counter, the sample may be biased. A larger sample size does not fix that.
• Getting the sample size effect backward. Larger random samples shrink the margin of error. A smaller margin of error means a more precise estimate.
• Treating the estimate as certain. "The proportion is 0.35" overstates what the sample shows. Look for words like "plausible," "estimate," or "likely."