Digital SAT · Math · Unit 3 · Problem Solving & Data

Lesson 3.6 · Samples, Generalizing & Margin of Error

On the test: 1–2 questions a test — words, not arithmetic: pick the option that says “plausible … between”, or the one about sample size

Big idea. We almost never ask everyone. We ask a random SAMPLE and use it to estimate the whole POPULATION. Three questions follow: How big is the estimate for everyone (scale it up)? Who does it apply to (only the population the sample was randomly picked from)? And how sure are we (the margin of error: the estimate ± the margin gives the plausible values — a bigger sample makes it narrower).

🎛 Move it · the Sampling Lab

Draw random samples from a population where the true value is 30 %. Every sample misses a little — that is what the margin of error measures. Then move the sample size and draw again.

sample size n = 100

With n = 100, the margin of error is about ±9.2%.

Every sample gives a slightly different estimate — that wobble is why a margin of error is reported. A bigger random sample wobbles less: 4 times the sample → half the margin. The estimate itself does not drift up or down because the sample is bigger.

🧑‍🏫 Rule 1 · Scale the sample upstep 1 of 5

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📋 The question · SAT · November 2023 · Math 800 Book Q1
In a random sample of 276 residents in a certain building, 9 residents reported that they take the train to work. If there are a total of 1,380 residents in this building, which of the following is the best estimate of the number of building residents who take the train to work?
1. A)9
2. B)45
3. C)276
4. D)1,095
the sample’s fraction
$\frac{9}{276}$
November 2023: 9 of the 276 sampled take the train.

🧑‍🏫 Rule 2 · (In the book’s section) “times as many” and “more than”step 1 of 3

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📋 The question · SAT · June 2024 · Math 800 Book Q4
A proposal for a construction project was included on a city election ballot. A TV news report stated that 8 times as many people voted in favor of the proposal as people who voted against it. An internet article reported that 24,500 more people voted in favor of the proposal than voted against it. Based on these data, how many people voted against the proposal?
(type the answer)
name them
$against = a, for = 8 a$
June 2024: 8 times as many voted for as against.

🧑‍🏫 Rule 3 · Generalize to the population sampled FROM — no furtherstep 1 of 2

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📋 The question · SAT · August US 2024 · Math 800 Book Q6
A district board is proposing a change to the time school starts for all high schools in the district. A sample of 353 high school students was selected at random from all high school students in the district; 300 of them said they did not approve. Which of the following is the largest population to which the results of the survey can be generalized?
1. A)All high school students in the district
2. B)The 353 students who were surveyed
3. C)The 300 students that responded that they did not approve of the proposed change
4. D)All high school students in the state
the population sampled from
$random from all district high schoolers$
August US 2024: the results speak for ALL high school students in the district (A) — not just the 353, and not the whole state.

🧑‍🏫 Rule 4 · A biased sample cannot be generalizedstep 1 of 3

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📋 The question · Gen Z Study · a biased sample
To estimate how many hours a week the town’s adults exercise, a researcher surveys 300 people leaving a gym. Why can the results NOT be generalized to all adults in the town?
1. A)300 people is too few
2. B)The sample was not selected at random from all adults in the town
3. C)Some people exercise at home
4. D)The survey asked about hours, not days
convenience sample
$gym-goers \neq = all adults$
People leaving a gym exercise more than the typical adult: the sample leans one way (bias). B.

🧑‍🏫 Rule 5 · Experiments and observational studiesstep 1 of 2

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📋 The question · Gen Z Study · experiment
200 volunteers were randomly assigned to two groups: one slept 8 hours, the other 5 hours, before a memory test. The 8-hour group scored much higher. Which is the most appropriate conclusion?
1. A)More sleep causes better memory scores for people like these volunteers
2. B)More sleep causes better memory scores for all people
3. C)There is an association, but no cause can be concluded
4. D)Memory causes people to sleep more
random ASSIGNMENT
$assigned 8 h or 5 h at random$
The researchers chose who slept how long, at random: any difference is caused by the sleep → cause and effect. But volunteers, not a random sample: only for people like them. A.
randomly ASSIGNED to groupsNOT randomly assignedrandomly SELECTED from a populationcause and effect ✓
for the whole population ✓association only
for the whole population ✓NOT randomly selected (volunteers, a gym, a website)cause and effect ✓
only for people like those studiedassociation only
only for those studied

🧑‍🏫 Rule 6 · Estimate ± margin = the plausible valuesstep 1 of 5

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📋 The question · SAT · March 2024 · Math 800 Book Q3
A machine fills bags with approximately 10 ounces of cranberries. To test the accuracy of the filling process, 148 bags were selected at random and weighed. Based on the sample, it is estimated that the average weight of all bags filled by the machine in an 8-hour period is 9.89 ounces, with an associated margin of error of 0.14 ounce. Which of the following is the best interpretation of this estimate?
1. A)Plausible values for the average weight of all bags filled by the machine are between 9.75 ounces and 10.03 ounces.
2. B)Plausible values for the average weight of all bags filled by the machine are less than 9.75 ounces or greater than 10.03 ounces.
3. C)The average weight of all bags filled by the machine is greater than 9.99 ounces.
4. D)The average weight of all bags filled by the machine is less than 9.99 ounces.
build the interval
$9.89 - 0.14 = 9.75, 9.89 + 0.14 = 10.03$
March 2024: plausible values for the TRUE average are between 9.75 and 10.03 ounces (A).

🧑‍🏫 Rule 7 · An interval in percent → an interval in countsstep 1 of 3

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📋 The question · SAT · March US 2026 (A) · Q17
An inspector checked a sample of 330 scales selected at random from a population of 8,000 scales to estimate what percentage were inaccurate. From the sample, the inspector estimated that 7% of the 8,000 scales were inaccurate, with an associated margin of error of 2.8%. Which is the most appropriate conclusion?
1. A)It is plausible that between 336 and 784 of the scales in the population are inaccurate.
2. B)It is plausible that fewer than 336 of the scales are inaccurate.
3. C)Exactly 560 of the scales are inaccurate.
4. D)It is plausible that more than 784 of the scales are inaccurate.
the percent interval
$7 \pm 2.8 = 4.2% to 9.8%$
March US 2026 (A): first the interval in percent.

🧑‍🏫 Rule 8 · Overlapping intervals decide nothingstep 1 of 1

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📋 The question · Gen Z Study · overlapping intervals
A poll estimates candidate A at 48% and candidate B at 45%, each with a margin of error of 3%. Which conclusion is best supported?
1. A)A will win
2. B)B will win
3. C)The poll does not show which candidate is ahead
4. D)Exactly 7% are undecided
two intervals
$A : 45 - 51, B : 42 - 48$
They overlap (45 to 48 is plausible for both), so either could really be ahead: C.

🧑‍🏫 Rule 9 · More confidence, wider intervalstep 1 of 1

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📋 The question · Gen Z Study · confidence level
From the same sample, an estimate is reported once with 95% confidence and once with 99% confidence. How do the margins of error compare?
1. A)The 99% margin is larger
2. B)The 99% margin is smaller
3. C)They are equal
4. D)It depends on the estimate
confidence level
$99% sure \Rightarrow wider$
To be more certain the interval catches the truth, it has to be wider: A. (The SAT nearly always uses 95 % and never makes you compute it.)

Real question 1SAT · October US 2024 · Math 800 Book Q10

A machine fills bags with approximately 29 ounces of walnuts. 169 bags were selected at random and weighed. The average weight of all bags filled in an 8-hour period is estimated to be 28.88 ounces, with an associated margin of error of 0.13 ounce. Which of the following is the best interpretation of this estimate?

🧑‍🏫 Rule 10 · A bigger random sample, a smaller marginstep 1 of 3

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📋 The question · SAT · August US 2024 · Math 800 Book Q9
The manager of a gym selected 139 members at random and estimates that 82% of members would keep paying if the price increased, with an associated margin of error of 6.39%. If the survey is repeated with a random sample of 278 members and the results are calculated in the same way, which is the most likely effect of using the larger sample?
1. A)The margin of error will be lower.
2. B)The margin of error will be higher.
3. C)The estimated percentage will be lower.
4. D)The estimated percentage will be higher.
the effect
$n : 139 \to 278 \Rightarrow margin ↓$
August US 2024: more data, less wobble from sample to sample: A.

🧑‍🏫 Rule 11 · Working backwards: which sample was bigger?step 1 of 4

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📋 The question · SAT · December 2023 · Math 800 Book Q5
A town manager selected 2 random samples of residents in the town to estimate the percentage of all residents that planned to vote in favor of a new business in the downtown area. Based on the first sample, he estimated that 75% planned to vote in favor, with an associated margin of error of 8.3%. Based on the second sample, he estimated 83%, with an associated margin of error of 6.7%. Assuming the margins of error were calculated the same way, which of the following best explains why the first sample obtained a larger margin of error than the second sample?
1. A)The first sample contained fewer residents than the second sample.
2. B)The first sample contained more residents than the second sample.
3. C)The first sample contained a lower percentage of residents that planned to vote in favor.
4. D)The first sample contained a higher percentage of residents that planned to vote in favor.
bigger margin ⇐ smaller sample
$8.3% > 6.7%$
December 2023: the first sample had the larger margin, so it had FEWER residents: A. The 75 % vs 83 % has nothing to do with it.

🧑‍🏫 Rule 12 · Four times the sample, half the marginstep 1 of 2

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the law
$margin \propto \frac{1}{n}$
The margin shrinks with the SQUARE ROOT of the sample size — doubling the sample does not halve it.

Trap · The three traps in these questions

1) “Exactly …” — an estimate is never exact. 2) “The margin is the percent” — 2.73% defective confuses the margin with the estimate. 3) “A bigger sample changes the estimate” — it only narrows the margin.

Take with you

▸Scale a random sample up: the same fraction (or percent) of the whole population.
▸Results generalize only to the population the sample was RANDOMLY selected from; a biased or self-chosen sample generalizes to no one.
▸Random ASSIGNMENT allows cause and effect; without it, only an association.
▸Estimate ± margin = the plausible values for the whole population — never “exactly”.
▸Turn a percent interval into counts by taking each end of the population.
▸Overlapping intervals cannot decide who is ahead; more confidence → a wider interval.
▸A bigger random sample gives a smaller margin (×4 the sample → ½ the margin); the estimate itself does not drift.

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