Math · Problem-Solving and Data Analysis

Inference from Sample Statistics and Margin of Error

A sample estimates a population quantity. A margin of error describes uncertainty around that estimate; a larger comparable random sample generally gives a narrower interval.

Written by Luke Finigan
2 min readUpdated

Build the interval around the estimate

A survey estimates support at 52% with a margin of error of 4 percentage points. Its interval runs from 48% to 56%: subtract and add 4 to the reported estimate.

This is uncertainty about the population proportion, not a claim that every individual answer is within 4% of something. The interval also does not prove support exceeds 50%, since values below 50% remain within it.

Identify the population the sample represents

A random sample of a city's registered voters can estimate that city's registered voters' opinions. It does not automatically represent all residents, all voters in a state, or people who were excluded from the sampling process.

Read how the sample was obtained. A large voluntary poll on a website can remain biased if its participants differ systematically from the target population.

Compare sample sizes under the same conditions

For comparable random samples at the same confidence level, a larger sample generally reduces margin of error. The approximate relationship is inverse to the square root of sample size: quadrupling the sample roughly halves the margin.

This comparison holds other features of the survey fixed. Increasing the sample size does not by itself repair a biased sampling method or a leading question.

Interpret estimates without promising certainty

If two intervals overlap, the reported intervals alone do not force a conclusion that the population values differ. Compare the possible ranges before asserting that one group definitely has the larger percentage.

Confidence describes the estimation procedure across repeated random samples. Do not reinterpret a reported confidence level as the percentage of respondents who answered accurately.

Check your understanding

Try it yourself

A poll estimates 38% with a margin of error of 3 percentage points. What is the reported interval?

Sample inference from sample statistics and margin of error questions

Inference from Sample Statistics and Margin of Error · Easy

A landscaping company studied a random sample of 24 lawns in a small town. The company found that 6 of these lawns contain fertilizer. If there are 288 lawns in the town, which of the following is the best estimate of the number of lawns that contain fertilizer?

Inference from Sample Statistics and Margin of Error · Easy

A district school board in a certain state is considering changing the starting time for all high schools in the district. A sample of 368 high school students was selected at random from all high school students in the district. The selected students were asked whether they approved of the proposed change, and 295 students responded that they did not approve. Which of the following is the largest population to which the survey results can be generalized?

Inference from Sample Statistics and Margin of Error · Medium

A group of 90 employees selected at random from all the employees at a company were surveyed about the number of books they read last year. From the data collected, it was estimated that the mean number of books employees at the company read last year is 8, with an associated margin of error of 1.4. Which of the following is the most appropriate conclusion?

Practice this skill

  1. Write the lower and upper endpoints with their units.
  2. Name the target population and sampling method.
  3. Separate uncertainty from sampling bias when reviewing a missed question.
Practice inference from sample statistics and margin of error

Print the inference from sample statistics and margin of error worksheet and worked answers.

Then use a mixed practice module to check whether you can choose this method among other question types.

Source and question labels

The bank label for this guide is "Inference from sample statistics and margin of error" in Problem-Solving and Data Analysis. The worked examples above are written for this guide.

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