inference from samples & margin of error

The SAT tests whether you understand how statistics from a random sample can be used to estimate population values—and how “margin of error” gives a likely range. Here you’ll see how to make valid conclusions, what you can and cannot infer, and how sample size, randomness, and error margin matter.

what is margin of error?

The margin of error tells you how much an estimated percent (or average) from a random sample might differ from the true value for the entire population. The true percent is probably within $+/-$ the margin of error.

Example:

In a poll, 30% of teens are “heavy texters” with a margin of error of 3%. That means the actual percent of all teens who are heavy texters is likely between 27% and 33%.

30\% \pm 3\% = \text{from } 27\% \text{ to } 33\%

SAT tip: The margin of error applies to the population percent or mean, not just those in the sample!

drawing valid conclusions

You can only generalize from a sample to a population if the sample is random and representative. If the margin of error is 3%, and the estimate is 40%, the true percent is probably between 37% and 43%. The larger the sample, the smaller the margin of error.

Example:

If a random sample of 20 out of 50 council members shows 6 support a bill, the best estimate for the city is:

\frac{6}{20} = 0.3 \text{ or } 30\% \text{ of 50} = 15

Multiply the percent by the whole group to estimate the number who support.

Try these sample-inference SAT questions:

Question 1

Texting behavior Talks on cell daily Does not talk on cell daily Total
Light 110 146 256
Medium 139 164 303
Heavy 166 74 240
Total 415 384 799
In a study of cell phone use, 799 randomly selected US teens were asked how often they talked on a cell phone and about their texting behavior. The data are summarized in the table above. Based on the data from the study, an estimate of the percent of US teens who are heavy texters is 30% and the associated margin of error is 3%. Which of the following is a correct statement based on the given margin of error?

Texting behavior	Talks on cell daily	Does not talk on cell daily	Total
Light	110	146	256
Medium	139	164	303
Heavy	166	74	240
Total	415	384	799

Question 2

A city has 50 city council members. A reporter polled a random sample of 20 city council members and found that 6 of those polled supported a specific bill. Based on the sample, which of the following is the best estimate of the number of city council members in the city who support the bill?

Question 3

A study was done on the weights of different types of fish in a pond. A random sample of fish were caught and marked in order to ensure that none were weighed more than once. The sample contained 150 largemouth bass, of which 30% weighed more than 2 pounds. Which of the following conclusions is best supported by the sample data?

✓

inference & margin of error mastered

You now know how to make inferences from samples, apply margin of error, and avoid classic SAT statistical traps.

Next Up

34: evaluating statistical claims from observational studies and experiments

master experiments, observational studies, and evaluating statistical claims