Estimate populations and margin of error

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Lesson progressPractice problems 0/3

Difficulty: Beginner
Estimated time: 28 minutes
Domains: Problem Solving and Data Analysis
Techniques: Sample-statisticsMargin-of-errorPlausible-intervalsPopulation-countsSample-size

What you’ll learn

Tell a number about the sample apart from the population value it estimates.
Turn an estimate and its margin of error into a plausible interval.
Read that interval as a claim about the population’s mean or percent.
Check whether a value or claim fits one or more plausible intervals.
Find a plausible count of people or things by typing % of straight into Desmos when the count isn’t quick to do in your head.
Explain why a bigger sample usually gives a more precise estimate.

Prerequisites

Read distributions and data displays0%
Read estimates and margins of error from a results table.

Why this matters on the SAT

Turn one estimate into a range

A sample gives you one number, an estimate for a much larger population. The margin of error tells you how far below or above that estimate the true population value could reasonably be. On the SAT, the margin of error is given to you. You won’t have to calculate it from raw data.

Solution to the example

Subtract the margin of error for the low end, and add it for the high end:

62\%-4\%=58\%

and

62\%+4\%=66\%.

So the true percentage is plausibly anywhere from $58\%$ to $66\%$ . Only $59\%$ lands in that range. $55\%$ is too low, and $67\%$ and $70\%$ are too high, so choice B is correct.

SAT example

A random sample of residents estimated that $62\%$ of all residents in a town support adding a bike lane. The estimate has a margin of error of $4$ percentage points.

Which choice could be the actual percentage of all residents in the town who support adding the bike lane?

A
$55\%$
B
$59\%$
C
$67\%$
D
$70\%$

Connect the sample to the population

A sample is the smaller group that actually got measured or surveyed. The population is the whole group the study wants to describe.

A number that sums up the sample, like its mean or its percent, is a sample statistic. The matching number for the whole population is a population parameter. The statistic is your estimate of the parameter.

Sample statistic	Population parameter it estimates
Mean of the sampled values	Mean for the whole population
Percent (or proportion) of the sample with a trait	Percent (or proportion) of the whole population with that trait

Say $80$ light bulbs are tested, and their mean life is $1{,}240$ hours. That $1{,}240$ is the sample mean. It’s an estimate of the mean life of all the bulbs the study is about. It doesn’t say every bulb lasts $1{,}240$ hours.

Percents work the same way. If $43\%$ of a sample picks an option, $43\%$ is the sample proportion. It estimates the population proportion, but it doesn’t promise that exactly $43\%$ of the population would pick it.

Check your understanding:

A sample of packages has a mean mass of $512$ grams. What does $512$ grams describe directly, and what does it estimate?

Common mistake:

An interval for a population mean is about one number: the population’s average. It doesn’t say where each individual value falls, and individual values can land below or above it. Before you read an interval, say what it’s about: “the mean for the population.”

Build and interpret the plausible interval

You already did the key move in the opening problem. If the estimate is $e$ and the margin of error is $m$ , the plausible interval is

\boxed{e-m\text{ to }e+m}.

You subtract and add the same margin, so the estimate always sits right in the middle.

Say a sample mean is $26.8$ minutes, with a margin of error of $1.7$ minutes:

26.8-1.7=25.1

and

26.8+1.7=28.5.

The conclusion to look for sounds like this: It is plausible that the population mean is between $25.1$ and $28.5$ minutes.

Plausible means reasonable, given the estimate and its margin of error. It doesn’t mean certain. On the SAT, a value outside the interval counts as not plausible, but that isn’t the same as impossible.

When the estimate is a percent, the margin is in percentage points, so you add and subtract it directly. For an estimate of $38\%$ with a margin of error of $5$ percentage points, the interval is

33\%\text{ to }43\%.

Common mistake:

It’s tempting to take $5\%$ of $38\%$ , which is $1.9$ points, and get $36.1\%$ to $39.9\%$ . That answers a percent-change question, which isn’t what a margin of error asks. The margin is $5$ points, so the interval is $38\%-5\%=33\%$ to $38\%+5\%=43\%$ . To check, make sure both endpoints sit exactly $5$ points from the estimate.

Check your understanding:

A sample estimates a population proportion as $0.47$ with a margin of error of $0.03$ . Give the plausible interval and name one value inside it.

Example: Compare two plausible intervals

When a table gives you two estimates, build both intervals before you judge any claim. If the intervals overlap, meaning they share at least one value, that shared value is plausible for both populations.

Worked example

A conservation group used separate samples to estimate the percentage of nesting boxes occupied at each of two sites.

Site	Estimated population percentage	Margin of error
A	$47\%$	$4$ percentage points
B	$54\%$	$3$ percentage points

Site A has an estimate of $47\%$ with a margin of error of $4$ percentage points.

Site B has an estimate of $54\%$ with a margin of error of $3$ percentage points.

Which conclusion is supported by the results?

A
The population percentage for Site A must be exactly $47\%$ .
B
$51\%$ is a plausible population percentage for both sites.
C
Every nesting box at both sites has an occupancy value between $43\%$ and $57\%$ .
D
The population percentage for Site B must be $7$ percentage points greater than the population percentage for Site A.

Step 1

Build the interval for Site A

Subtract and add Site A’s margin of $4$ points:

47\%-4\%=43\%

and

47\%+4\%=51\%.

Site A’s plausible interval is $43\%$ to $51\%$ .

Step 2

Build the interval for Site B

Now Site B, with a margin of $3$ points:

54\%-3\%=51\%

and

54\%+3\%=57\%.

Site B’s plausible interval is $51\%$ to $57\%$ .

Step 3

Use the overlap

Line the two intervals up:

\text{Site A: }43\%\text{ to }51\%

\text{Site B: }51\%\text{ to }57\%.

They meet at exactly $51\%$ , the top of A’s interval and the bottom of B’s. So $51\%$ is plausible for both sites, and choice B is correct.

The other choices overclaim. A says “must be exactly,” which an estimate can’t promise. C talks about every nesting box instead of the population percentage. And D takes the $7$ -point gap between the estimates as fact, but the true percentages aren’t guaranteed to be $7$ points apart.

Check your understanding:

Do the overlapping intervals prove that the two population percentages are equal? Explain.

Common mistake:

The gap between two estimates isn’t automatically the gap between the populations. Here the estimates differ by $7$ points, but the true percentages could both be $51\%$ , a gap of $0$ , or as far apart as $43\%$ and $57\%$ , a gap of $14$ . Build both intervals and test the exact claim against their endpoints.

Use intervals to estimate population counts

Sometimes the SAT asks for a plausible number of people or things instead of a percent. Build the percent interval first. Then turn the endpoint you need into a count of the population.

Say a town has $10{,}000$ households. A sample estimates that $36\%$ of them use solar power, with a margin of error of $3$ percentage points.

The plausible percent interval is

33\%\text{ to }39\%.

Picking the endpoints is your job. When the counts aren’t quick to do in your head, turning endpoints into counts is Desmos’s job, and you can type each one exactly as it reads, percent sign and all:

Enter 33% of 10000.
Enter 39% of 10000.
Read the two outputs as numbers of households.

Desmos returns $3{,}300$ and $3{,}900$ , so the plausible number of households with solar power runs from $3{,}300$ to $3{,}900$ . Typing % of means you don’t have to turn the percent into a decimal or work out the multiplication yourself.

Which endpoint you need depends on the question. The greatest plausible count comes from the upper endpoint, and the least plausible count comes from the lower one.

Try it yourself:

A population has $6{,}000$ members. An estimated proportion is $41\%$ , with a margin of error of $2$ percentage points. Before you reveal the answer, find the greatest plausible number of members with the trait.

Check your understanding:

What is the greatest plausible number of members in the situation above?

Calculator loads as you approach

Each line is one endpoint, typed as a percent of the population. Reset restores the example.

Connect sample size to precision

A smaller margin of error makes a narrower interval. That makes the estimate more precise: it pins the population value down to a tighter range.

So what shrinks the margin? When comparable studies use the same method on the same kind of population, a larger sample generally gives a smaller margin of error. A bigger sample holds more information about the population, so the estimate tends to change less from one sample to the next.

Sample size	General effect on margin of error	General effect on precision
Larger	Smaller	Greater
Smaller	Larger	Lower

Notice the word generally. It’s a tendency, not a guarantee. And sample size only tells you about the margin. It doesn’t tell you whether the estimate itself will come out larger or smaller, and it doesn’t tell you the range or standard deviation of the sampled data.

Sample size isn’t the only factor. When the values in a population vary a lot, its mean is harder to estimate precisely. You won’t calculate a margin from either one. Go with whichever factor the question names.

Check your understanding:

Two surveys use the same method and sample from the same population. Survey P samples $900$ people, and Survey Q samples $300$ people. Which survey will generally have the smaller margin of error? What can’t you predict from sample size alone?

Common mistake:

It’s easy to think a bigger sample pushes the estimate up, as if more people meant a bigger number. It doesn’t. A bigger sample generally narrows the range around the estimate without moving it up or down. When sample sizes differ and the question gives you nothing else, compare margins of error or precision, not the estimates.

Keep nearby statistical questions separate

Everything so far answers one kind of question: what a sample mean or percent, plus its margin of error, says about the population, and how sample size affects precision. A few nearby questions sound similar but ask something else.

If the question asks who a result can speak for, or whether the sampling method gives a representative sample, that’s about Decide when a sample supports generalization.

If it asks whether a treatment caused an outcome, that’s about Decide when a study supports causation.

If it asks how spread out the data values themselves are, that’s about Reason with range and standard deviation. A margin of error measures how uncertain an estimate is, not how spread out the individual values are.

Check your understanding:

What kind of question is each one? (1) Find the interval for $48\%\pm3$ percentage points. (2) Decide whether surveying only gym members can represent every adult in a city. (3) Compare how spread out two lists are.

Practice problems

Each problem takes one more step: a mean interval, then a count, then a total across two groups.

Find a plausible mean interval

Practice problem

A random sample of rechargeable batteries had a mean operating time of $14.8$ hours. The estimate of the mean operating time for all batteries of this type has a margin of error of $1.1$ hours.

Which conclusion is most appropriate?

Calculator loads as you approach

Find both endpoints, then pick the wording about the mean.

Find the greatest plausible count

Practice problem

A town has $7{,}200$ registered bicycles. A random sample is used to estimate that $43\%$ of all registered bicycles have front baskets, with a margin of error of $5$ percentage points.

What is the greatest plausible number of registered bicycles in the town that have front baskets?

Your answer

Calculator loads as you approach

Pick the endpoint first, then type % of the town’s bicycles.

Combine intervals for two groups

Practice problem

A college has $2{,}400$ commuting students and $3{,}600$ students who live on campus. Separate samples estimate the percentage of each group that uses a meal plan.

Group	Estimated percentage	Margin of error
Commuting students	$38\%$	$4$ percentage points
Students living on campus	$46\%$	$3$ percentage points

Which choice gives the range of plausible values for the total number of college students who use a meal plan?

Calculator loads as you approach

One expression for the low total, one for the high total.

Finish the lesson

3 practice examples left

Finish the remaining questions correctly to complete this lesson.

Quick recap

A sample mean or percent is a sample statistic. It estimates the matching population parameter.
With estimate $e$ and margin of error $m$ , the plausible interval runs from $e-m$ to $e+m$ .
An interval for a mean is about the population mean, not every individual value.
For percent estimates, add and subtract the percentage points directly.
Overlapping intervals mean a shared value is plausible for both. They don’t prove the two parameters are equal.
Build and read intervals by reasoning. For a plausible count, pick the endpoint. If the count isn’t quick to do in your head, type one % of expression into Desmos instead of converting to a decimal and working it out by hand.
A larger sample generally means a smaller margin of error and a more precise estimate. It doesn’t tell you whether the estimate will rise or fall.

Next lesson

Decide when a sample supports generalization

Judge whether a sampling method supports extending a result to a named population.

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307 SAT questions use what this lesson teaches. Practice a few in a study session at the difficulty you choose.

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