Single Step SAT Prep

Digital SAT Math · Problem-Solving & Data Analysis

Inference from sample data

A sample hands you one number — a percent, a mean, a count — and a margin of error. The skill is not computing that margin from a formula (the Digital SAT never asks you to). The skill is reading the number as an interval of plausible values for the population, and knowing what makes that interval narrower.

On the test

DomainProblem-Solving and Data Analysis (score report)
CB skillInference from sample statistics and margin of error
What it looks likeA short survey or study result: sample estimate, often a margin of error, sometimes a population size to scale to
Often asked“Which is a plausible value…?”, “Which interval…?”, “Which conclusion is best supported?”, “About how many of the whole population…?”
FormatMultiple choice and student-produced response
CalculatorAllowed; arithmetic is usually trivial (add/subtract the margin, multiply a proportion by NN)

Recognition cues: random sample, margin of error, percentage points, plausible value, estimate, based on the sample, all … in the city/colony/company.

Pattern recognition

Four shapes cover nearly every item. Name the shape before you touch a number.

  1. Build the interval — estimate and margin given; write estimate±margin\text{estimate} \pm \text{margin}.
  2. Read a plausible value — a candidate is inside the interval (supported) or outside it (not supported by this sample).
  3. Scale to the population — sample proportion ×\times population size. If a margin is given, scale the interval ends, not only the point estimate.
  4. What narrows the margin — larger random sample → narrower interval; the centre (the estimate) does not move just because nn grew.

Method

  1. Write the interval. Subtract the margin from the estimate; add the margin to the estimate. Units stay the units of the estimate (percent, minutes, kilograms, people).
  2. Say it in population language. “Any value from LL to UU is a plausible value for the [population percent / population mean].” Do not say “every individual falls between LL and UU.”
  3. Test each claim against the interval. Inside → can be supported. Outside → not supported by this sample (which is not the same as “impossible forever”).
  4. Scale carefully when NN is given. p^=count in samplen\hat{p} = \frac{\text{count in sample}}{n}, then p^×N\hat{p}\times N for a population count. With a margin in percentage points, convert to a proportion margin if needed and scale both ends.
  5. Compare two studies by overlap and by width. Overlapping intervals can both be right; the larger random sample is the more precise (narrower margin). The centre only moves if the estimate moves — growing nn alone does not relocate the centre.
Words in the stemWhat to do
margin of error of mminterval is estimate ±m\pm m
plausible value / most plausiblecheck membership in the interval
NOT a plausible valuethe choice that sits outside
about how many of the NN…proportion ×N\times N
larger sample / more precisenarrower margin of error; centre unchanged unless the estimate changes
all individuals / every person in the samplewrong target — the interval is for the parameter
respondents who made an errorwrong meaning of “margin of error”

Worked example 1 — build the interval and test a claim

Stem. A random sample of voters finds that 52% support a transit measure, with a margin of error of 3 percentage points. (a) What is the interval of plausible values for the percent of all voters who support the measure? (b) Is 48% a plausible value? Is 51%?

Step 1 — write the interval.

52%−3%=49%,52%+3%=55%52\% - 3\% = 49\%, \qquad 52\% + 3\% = 55\%

So the interval is 49% to 55%.

Step 2 — test the candidates. 48% is below 49%, so it is not a plausible value based on this sample. 51% sits inside 49–55, so it is plausible.

Check. The centre is the sample estimate (52%), and the half-width is exactly the stated margin (3). Answer: (a) 49% to 55%; (b) 48% no, 51% yes.

Trap watch. Reporting 52% to 55% (adding only) or 49% to 52% (subtracting only) builds a one-sided stub. Treating 48% as “impossible” rather than “not supported here” overstates what a margin of error can do.

Worked example 2 — scale a sample, then scale the interval

Stem. A city has 8,000 households. A random sample of 200 households finds that 50 plan to attend a recycling workshop. The margin of error for the sample percent is 4 percentage points. About how many of the 8,000 households plan to attend, and what is a plausible range for that count?

Step 1 — sample proportion.

p^=50200=0.25=25%\hat{p} = \frac{50}{200} = 0.25 = 25\%

Step 2 — point estimate for the city.

0.25×8,000=2,000 households0.25 \times 8{,}000 = 2{,}000 \text{ households}

Step 3 — interval on the percent, then scale both ends.

25%±4%  ⇒  21% to 29%25\% \pm 4\% \;\Rightarrow\; 21\% \text{ to } 29\% 0.21×8,000=1,680,0.29×8,000=2,3200.21 \times 8{,}000 = 1{,}680, \qquad 0.29 \times 8{,}000 = 2{,}320

Check. The point estimate 2,000 sits at the centre of 1,680–2,320. Answer: about 2,000 households; a plausible range is 1,680 to 2,320.

Trap watch. Scaling only the point estimate and then adding ±4\pm 4 households (or ±4%\pm 4\% of 200) ignores that the margin is in percentage points on the percent, not raw people. Another common miss: using 50 as if it were already the city-wide count.

Worked example 3 — two polls, overlap, and what larger nn does

Stem. Two polls ask the same question of random samples of city residents.

  • Poll A (n=300n = 300): 62% support a new park, margin of error 5 percentage points.
  • Poll B (n=1,200n = 1{,}200): 58% support, margin of error 3 percentage points.

(a) Write both intervals. (b) Could the true support be 59%? (c) Which poll is more precise?

Step 1 — intervals.

Poll A: 62±5  ⇒  57% to 67%\text{Poll A: } 62 \pm 5 \;\Rightarrow\; 57\% \text{ to } 67\% Poll B: 58±3  ⇒  55% to 61%\text{Poll B: } 58 \pm 3 \;\Rightarrow\; 55\% \text{ to } 61\%

Step 2 — overlap. 59% lies in both intervals, so both polls can support 59% as a plausible population value. The polls are not forced to contradict each other just because the point estimates differ.

Step 3 — precision. Poll B used the larger sample and reports the smaller margin, so Poll B is more precise. The centre of each interval is still that poll’s own estimate; growing nn narrows the band without, by itself, inventing a new centre.

Check. The overlap band includes 57–61; 59 is inside it. Answer: (a) 57–67 and 55–61; (b) yes; (c) Poll B.

Trap watch. “The true value must be exactly 62% because Poll A said 62%” treats a sample statistic as exact. “59% is impossible because the two point estimates disagree” ignores interval overlap. “Poll A is more precise because 62 > 58” confuses the size of the estimate with the size of the margin.

Practice

Answer before you open the explanation. Two items are student-produced response (type the number, no choices), matching the real test. Every wrong choice below is a specific error with a name — when you miss one, log the name, not the item. No figures: this skill is almost always pure text on the Digital SAT.

12 questions — 10 multiple choice, 2 student-produced response. Every wrong choice has its own explanation.

Question 1 Warm-up

A poll of a random sample of city residents finds that 58% support a new branch library, with a margin of error of 3 percentage points. Which of the following is the interval of plausible values for the percent of all city residents who support the library?

Show the answer Choice B

Why it is right

The margin of error is applied symmetrically around the sample estimate: 58% − 3% = 55% and 58% + 3% = 61%. So any population support percent from 55% to 61% is a plausible value based on this sample. The interval is about all city residents, not about a one-sided half of the margin.

Why each other choice fails

Choice A
Subtracts the margin on the low side only and stops at the sample estimate, producing a half-interval 55% to 58% instead of the full ±3 band.
Choice C
Adds the margin on the high side only, producing 58% to 61% and ignoring that the same 3 points also extend below 58%.
Choice D
Anchors the low end at the margin itself (3%) rather than at 58% − 3%, which is not how a margin of error interval is built.

Question 2 Standard

A random sample of 90 parcels from a delivery network has a mean weight of 4.8 kilograms and a margin of error of 0.3 kilograms. Which of the following is a plausible value for the mean weight of all parcels in the network?

Show the answer Choice A

Why it is right

The interval of plausible values for the population mean is 4.8 ± 0.3, which is 4.5 kg to 5.1 kg. Among the choices, only 4.6 lies inside that interval, so only 4.6 is a plausible value for the mean weight of all parcels in the network. The sample mean 4.8 is the centre; the margin sets how far a population mean can sit from that centre.

Why each other choice fails

Choice B
4.2 is below the lower end of 4.5, so it sits outside the interval and is not a plausible population mean based on this sample.
Choice C
5.2 is above the upper end of 5.1, so it is outside the interval even though it is only 0.1 kg past the edge.
Choice D
0.3 is the margin of error itself, not a candidate for the mean weight of all parcels in the network.

Question 3 Standard

A random sample of 250 of the 5,000 employees at a company finds that 80 employees bike to work. Based on the sample, about how many of the 5,000 employees bike to work?

Show the answer Choice C

Why it is right

The sample proportion is 80/250 = 0.32. Scaling that proportion to the full company gives 0.32 × 5,000 = 1,600 employees. The sample is treated as representative of the 5,000, so the estimate for the population count is the sample rate times the population size, not the raw sample count.

Why each other choice fails

Choice A
Reports the number of bikers in the sample (80) without scaling up to the 5,000 employees in the company.
Choice B
Reports the sample size (250) rather than an estimated count of bikers among all 5,000 employees.
Choice D
Uses 0.40 × 5,000 = 2,000, as if 100 of 250 had biked, or roughly rounds 80/250 up to two-fifths without computing 0.32.

Question 4 Standard

Two research teams use the same random-sampling method to estimate the percent of adults in a city who own a bicycle. Team R surveys 150 adults; Team S surveys 1,500 adults. Which of the following statements is most likely true?

Show the answer Choice B

Why it is right

With the sampling method held fixed, a larger random sample produces a smaller margin of error. Team S surveys ten times as many adults as Team R, so Team S's estimate of the city-wide percent is expected to be more precise — a narrower interval around its sample result. The method being the same does not cancel the effect of sample size.

Why each other choice fails

Choice A
Reverses the sample-size rule: the smaller sample (Team R) is the one expected to have the larger margin of error, not the smaller one.
Choice C
Same method does not mean same precision. Sample size still drives the width of the margin; 1,500 is much larger than 150.
Choice D
Two random samples of different sizes can easily produce different point estimates; nothing forces the two percents to match exactly.

Question 5 Standard

An ecologist selected a random sample of 40 songbirds from a preserve and found a mean wingspan of 18.6 centimetres with a margin of error of 1.4 centimetres. Which of the following is the best interpretation of these findings?

Show the answer Choice D

Why it is right

The sample mean 18.6 estimates the population mean for the preserve, and the margin of error builds the interval 18.6 ± 1.4, or 17.2 to 20.0 centimetres. That interval is a set of plausible values for the population mean, not a claim that every individual bird falls inside those bounds, and not a claim that 18.6 is exact. Choice D is the only sentence that names the right target (the population mean) and the right level of certainty (plausible, not exact).

Why each other choice fails

Choice A
Applies the interval to every individual in the sample. Individual wingspans vary; the margin of error is about the mean, not about each bird's measurement.
Choice B
Applies the interval to every individual in the preserve. The claim is about the mean wingspan of all songbirds, not about each bird's wingspan.
Choice C
Treats the sample mean as exact for the population and ignores the margin of error entirely.

Question 6 Harder

Each person in a random sample of 200 city residents was asked whether they favor a new bike lane. Of those surveyed, 34% were in favor. If the margin of error associated with the survey results is 6 percentage points, which of the following is most plausible?

Show the answer Choice A

Why it is right

The interval of plausible values for the city-wide percent is 34% ± 6%, or 28% to 40%. Among the choices that talk about all city residents, 38% lies inside that interval and 42% does not, so 38% is the most plausible population claim. Choices about the surveyed residents misread the sample: the sample percent is already known to be 34%, and the margin is not a lie rate.

Why each other choice fails

Choice B
Reads the margin of error as the share of respondents who lied. The margin is sampling uncertainty about the population percent, not a count of dishonest answers.
Choice C
States a percent of the surveyed residents, but the sample result is already known: 34% of those surveyed were in favor, not 28%.
Choice D
42% is above the upper end of the interval 28% to 40%, so it is not a plausible value for the city-wide percent based on this sample.

Question 7 Harder Student-produced response

A study estimates that 2,400 residents of a town use the public library each month, with a margin of error of 180 residents. What is the lower end of the interval of plausible values for the number of residents who use the library each month?

Show the answer 2220

Why it is right

The interval of plausible values is the estimate plus or minus the margin of error: 2,400 ± 180. The lower end is 2,400 − 180 = 2,220 residents. That is the smallest monthly user count still treated as plausible based on the study's estimate and margin.

Answers students type instead

180
Reports the margin of error alone, without combining it with the 2,400 estimate.
2400
Reports the point estimate instead of subtracting the margin to reach the lower end of the interval.
2580
Adds the margin instead of subtracting it, producing the upper end (2,400 + 180) rather than the lower end.

Question 8 Harder

A poll estimates that 47% of voters support a candidate, with a margin of error of 4 percentage points. Which of the following is NOT a plausible value for the true percentage of voters who support the candidate?

Show the answer Choice D

Why it is right

The interval of plausible values is 47% ± 4%, or 43% to 51%. Values 44%, 47% and 51% all sit inside that closed interval (51% is exactly the upper end). 53% is 2 points above 51%, so it is not a plausible value for the true support percent based on this poll.

Why each other choice fails

Choice A
44% is above the lower end of 43% and below 51%, so it is inside the interval and is a plausible value.
Choice B
47% is the sample estimate itself and is always the centre of the interval, so it is a plausible value for the population percent.
Choice C
51% is exactly 47% + 4%, the upper end of the interval, so it is still a plausible value.

Question 9 Harder

Two polls estimate the percent of city residents who support a new park. Poll A surveyed 400 residents and found 62% support with a margin of error of 5 percentage points. Poll B surveyed 1,600 residents and found 58% support with a margin of error of 3 percentage points. Which of the following statements is best supported by these results?

Show the answer Choice C

Why it is right

Poll A's interval is 62% ± 5%, or 57% to 67%. Poll B's interval is 58% ± 3%, or 55% to 61%. The value 59% sits in both intervals, so both polls can treat 59% as a plausible population support percent. Overlapping intervals mean the polls need not contradict each other even though their point estimates differ. Poll B's larger sample also makes it the more precise of the two, but the supported claim in the key is the overlap claim.

Why each other choice fails

Choice A
Treats a sample percent as exact for the whole city. The margin of error exists precisely because 62% is an estimate, not a certainty.
Choice B
Confuses disagreeing point estimates with non-overlapping intervals. 59% is inside both 57–67 and 55–61, so both polls can support it.
Choice D
Confuses the size of the point estimate with precision. Precision is about the margin of error (and sample size); Poll B has the smaller margin and the larger sample.

Question 10 Hardest

A polling firm wants a narrower margin of error for its estimate of the percent of voters who favor a bond measure, without changing how voters are randomly selected. Which of the following changes is most likely to produce a narrower margin of error?

Show the answer Choice A

Why it is right

Holding the random-selection method fixed, the reliable way to shrink a margin of error is to grow the sample size. A larger random sample produces a tighter interval around the sample estimate. Raising a confidence level widens the interval (you buy more confidence with a broader band), a smaller sample widens the margin, and simply omitting the margin from a report does not make the estimate more precise.

Why each other choice fails

Choice B
A higher confidence level widens the interval, not narrows it. The trap is assuming that 'more confidence' means 'a tighter band.'
Choice C
A smaller sample increases sampling uncertainty and typically produces a larger margin of error, the opposite of the firm's goal.
Choice D
Hiding the margin of error does not change the sampling uncertainty. The estimate is no more precise just because the margin is left off the page.

Question 11 Hardest

A random sample of adults in County M finds that 55% support a proposed trail, with a margin of error of 3 percentage points. Which of the following is a correct interpretation of the results?

Show the answer Choice B

Why it is right

The interval of plausible values for the county-wide support percent is 55% ± 3%, or 52% to 58%. The value 53% lies inside that interval, so it is a correct, carefully worded claim about the population of all adults in County M. The other choices misuse the margin as a lie rate, overstate outside values as impossible, or reapply the interval to the sample whose percent is already known to be 55%.

Why each other choice fails

Choice A
Treats the margin of error as the share of respondents who answered incorrectly. The margin measures sampling uncertainty about the population percent, not respondent mistakes.
Choice C
Overstates what an interval can do: a value outside 52–58 is not plausible based on this sample, but that is not the same as proving the true percent cannot be 50%.
Choice D
Applies the interval to the sample. The sample percent is already known (55%); the interval is a claim about all adults in County M, not about re-describing the sample.

Question 12 Hardest Student-produced response

In a random sample of 400 of a city's 20,000 households, 140 reported composting food waste. The margin of error for the sample percent is 4 percentage points. Based on the sample, what is the greatest number of the city's 20,000 households that is a plausible value for the number that compost food waste?

Show the answer 7800

Why it is right

The sample percent is 140/400 = 0.35 = 35%. With a margin of 4 percentage points, the upper end of the interval for the city-wide percent is 35% + 4% = 39%. Scaling that upper end to the city gives 0.39 × 20,000 = 7,800 households. That is the greatest city-wide composting count still treated as plausible under the given margin.

Answers students type instead

140
Reports the sample count of composting households without scaling to the city's 20,000 households.
6200
Uses the lower end of the interval (31% of 20,000) instead of the upper end the question asks for.
7000
Scales only the point estimate 0.35 × 20,000 = 7,000 and never adds the 4 percentage-point margin to reach the upper end.
7600
Uses 38% of 20,000, as if the upper end were 35% + 3 or a one-point slip off 39%.

Common mistakes

  1. Reading the margin as respondents’ error — treating ±3%\pm 3\% as “3% of people answered wrong.” The margin is sampling uncertainty about the population parameter.
  2. Applying the interval to the sample, not the population — the sample percent is already known; the interval is a claim about all members of the population the sample was drawn from.
  3. Treating every individual as inside the interval — a mean of 34±334 \pm 3 minutes does not mean every person commutes between 31 and 37 minutes.
  4. Treating values outside the interval as impossible — outside means “not plausible based on this sample,” not “ruled out forever.”
  5. Assuming a higher confidence level narrows the interval — higher confidence widens the band; a larger sample is what narrows it.
  6. Scaling the sample count without the population size — reporting 80 bikers from a sample of 250 when the question asked about 5,000 employees.
  7. Adding the margin only on one side — writing 52% to 55% when the margin is ±3\pm 3.
  8. Using the sample size as the population size when scaling a proportion.
  9. Declaring two polls contradictory whenever their point estimates differ, even though their intervals overlap.
  10. Believing a larger sample moves the centre — larger nn shrinks the margin; the estimate only moves if the new data actually produce a new statistic.

FAQ

Do I ever have to compute a margin of error from a formula? No. The Digital SAT gives you the margin (or asks you to reason about what would make it smaller). There is no z∗σ/nz^*\sigma/\sqrt{n} work on this test.

Is the sample percent itself a plausible value for the population? Yes — it is the centre of the interval, so it always sits inside estimate±margin\text{estimate} \pm \text{margin}. Choices that treat the sample value as exact for the population are still wrong; “plausible” is not “certain.”

What if two intervals do not overlap? Then the two samples disagree more sharply: there is no shared band of population values both support. The test still will not ask you to run a formal significance test — just to notice the lack of overlap.

How is this different from evaluating statistical claims? This page is about reading an estimate and its margin. Evaluating statistical claims is about design: who was randomly selected, whether a treatment was randomly assigned, and which causal or generalising sentence the design can support. Keep bias and causation off this skill’s answer choices.

How do I enter an SPR answer? Type the number the last line asks for — a lower bound, an upper bound, or a scaled count — without units or a percent sign unless the stem tells you to include one. Commas are optional in the grid.

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Editorial team · last reviewed 2026-08-12