One of the two rarest Math types (1.3% of the bank) and one of the highest returns per hour, because it is almost pure vocabulary. Covers reading a confidence interval as an estimate plus or minus a margin, why a larger sample narrows the interval, what a margin of error does and does not mean, and the rule that conclusions extend only to the population actually sampled — with six worked examples, four traps and three checks.
Subject: SAT Prep · 61 slides · applied lesson
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Title
SAT Math · Type 18 of 19
1.3% of the question bank — 22 of 1675 questions
Objectives
A survey is taken, a statistic is reported, and a margin of error is attached to it. Every question on this type is about what that combination does and does not license you to say. There is almost no arithmetic: the interval is an addition and a subtraction, and everything else is knowing three rules about samples.
Two sentences cover nearly the whole type: a bigger sample narrows the interval, and conclusions stop at the edge of the population you sampled.
Naruhodo Tutoring — SAT Math question bank export (sat-question-index.json) — 22 tagged questions of this type in the site's bank
Section
Section 1
Concept
A survey is taken, a statistic is reported, and a margin of error is attached to it. Every question on this type is about what that combination does and does not license you to say. There is almost no arithmetic: the interval is an addition and a subtraction, and everything else is knowing three rules about samples.
You will see it phrased in these ways:
The third phrasing is the one that separates a prepared student from an unprepared one. The answer is always: only the population that was actually sampled.
Picture it
The same three-panel card as the survey deck, so the shorthand carries over: what identifies it, what you write first, and what is built to catch you.
Figure (svg): Sample statistics and margin of error: the tell, the move, and the trap
Every panel here is a sentence rather than a technique. That is the character of the type, and it is why an hour of vocabulary work is worth more than an hour of practice questions.
Prediction
The most misunderstood term in the type.
Predict first
A survey reports 42 per cent with a margin of error of 3 per cent. What does the 3 per cent describe?
Correct: The uncertainty in the estimate due to sampling only part of the population
Why: A margin of error quantifies how much the sample estimate might differ from the true population value, purely because a sample is not the whole population. It has nothing to do with mistakes, misreadings or bad data — the word error is misleading in ordinary English. It says the true value plausibly lies between 39 and 45 per cent.
Concept
Three shapes, and only the first involves arithmetic.
| variant | what it wants | the move |
|---|---|---|
| Build the interval | the plausible range for the true value | estimate plus or minus the margin |
| Sample size | the effect of surveying more people | a larger sample narrows the interval |
| Generalisation | which group the result applies to | only the population actually sampled |
The third variant is the most common and needs no numbers at all. It is answered by reading who was surveyed.
Definition probe
The vocabulary decides everything else.
Sort into buckets
Which is each of these?
Discrimination
Six changes to a survey.
Sort into buckets
Does the margin of error grow, shrink, or stay about the same?
Warm-up
Try it, then read what it means.
Discussion prompt
A poll of 800 voters finds 46 per cent support a proposal, with a margin of error of 3.5 per cent. What is the plausible range for the true level of support?
Hint: Add and subtract the margin.
Answer:
Between 42.5 per cent and 49.5 per cent. The interval is the estimate plus or minus the margin: 46 minus 3.5 and 46 plus 3.5.
What it means: the true level of support among all voters plausibly lies somewhere in that range.
What it does not mean: it is not a claim that 42.5 per cent of people said one thing. The interval is about the unknown population value, not about the sample.
A useful consequence: since the whole interval sits below 50 per cent, this poll suggests the proposal lacks majority support — which is the kind of conclusion the SAT asks you to draw.
Pattern
Three steps, and two of them are reading.
Identify who was actually sampled, and write it down.
Why: Every conclusion question is decided by this. The result applies to that group and to no wider one.
If a margin is given, build the interval: estimate minus margin to estimate plus margin.
Why: This is the only arithmetic in the type, and it is one addition and one subtraction.
Check the conclusion against both: does it stay inside the interval, and inside the sampled population?
Why: The two standard traps are a claim that is too precise and a claim that is too broad, and this checks for both.
Step 1 first, always. A perfectly computed interval attached to the wrong population is still the wrong answer.
Section
Section 2
Concept
The plausible range for the true population value runs from the estimate minus the margin to the estimate plus the margin.
| estimate | margin | interval |
|---|---|---|
| 46 per cent | 3.5 per cent | 42.5 to 49.5 per cent |
| 120 minutes | 8 minutes | 112 to 128 minutes |
| 3.2 children | 0.4 | 2.8 to 3.6 |
| 70 per cent | 2 per cent | 68 to 72 per cent |
The whole arithmetic of this type is in that table. Everything else is knowing what the interval means.
Concept
More respondents means more information, and therefore a smaller margin of error.
That last point surprises people and is worth carrying: a national poll and a town poll of the same size have about the same margin.
Prediction
One addition and one subtraction.
Predict first
A study estimates the mean commute at 34 minutes with a margin of error of 2.5 minutes. What is the plausible range?
Correct: 31.5 to 36.5 minutes
Why: Subtract and add the margin: 34 minus 2.5 is 31.5, and 34 plus 2.5 is 36.5. The interval is centred on the estimate and has a full width of twice the margin, which is 5 minutes. The other choices each use only one side of the interval, which halves it.
Concept
A result describes the group the sample was drawn from, and says nothing about any wider group.
On a conclusion question, find the sentence naming who was surveyed and reject every choice mentioning a wider group.
Prediction
Direction only — the SAT never asks for the formula.
Predict first
A survey of 400 people has a margin of error of 5 per cent. If 1,600 people were surveyed instead, the margin would be
Correct: smaller
Why: A larger sample provides more information about the population, so the estimate is more precise and the margin shrinks. Quadrupling the sample roughly halves the margin, though the SAT only tests the direction. Note that the margin depends on the sample size and not on how large the population is.
Concept
It quantifies the uncertainty from measuring part of a population rather than all of it.
The word error is misleading. A better name would be a margin of sampling variability, and thinking of it that way prevents most of the misreadings.
Concept
If the sample is not representative, no margin of error and no sample size makes the result generalisable.
So a question describing a self-selected sample has a correct answer that criticises the method, not one that quotes a narrower interval.
Prediction
Find the sentence naming who was surveyed.
Predict first
A survey randomly selected 300 students from one high school. The results can be generalised to
Correct: students at that high school
Why: Random selection licenses generalisation to the population the sample was drawn FROM, which here is the students of that one school. Nothing about the study speaks to other schools, whose students might differ in every relevant way. Each wider choice claims more than the sampling frame supports.
Concept
If two confidence intervals overlap, the data does not establish that the two true values differ.
This is a favourite SAT conclusion question, and the correct answer usually says the difference cannot be established.
Concept
A confidence interval describes a population average or proportion, not any single member.
That confusion between the precision of an average and the spread of the data is a distractor the SAT builds deliberately.
Prediction
Does the data establish a difference?
Predict first
Group A is estimated at 52 per cent with a margin of 4, and Group B at 55 per cent with a margin of 4. What can be concluded?
Correct: The data does not establish that the groups differ
Why: The intervals are 48 to 56 and 51 to 59, which overlap substantially. Since the true values could both be, say, 54, the survey has not shown a real difference. Note that this is not the same as showing the groups are equal — failing to establish a difference is not evidence of sameness.
Two truths and a lie
Three of these statements about sampling are correct. The one left standing is false.
Eliminate the wrong options
Which statement is FALSE?
Survives elimination: c
Why: The margin of error depends on the SAMPLE size, not on the population size. A properly drawn sample of 1,000 gives about the same precision whether the population is 50,000 or 50 million. This surprises people, and it is why national polls can use samples of about a thousand people. The only thing that matters is how many you asked and whether they were selected properly.
Check
One addition and one subtraction.
Check your understanding
A survey estimates that 57 per cent of a company's employees use public transport, with a margin of error of 3.2 per cent. Which interval is plausible for the true percentage?
Answer: A
Why: Subtract and add the margin: 57 minus 3.2 is 53.8, and 57 plus 3.2 is 60.2. The interval is centred on the estimate with a full width of 6.4, which is twice the margin.
The width check settles it instantly: the correct interval must be exactly twice the margin wide, and only choice A is 6.4 points across.
Section
Section 3
Worked example
A poll of 1,200 residents estimates that 38 per cent support a new library, with a margin of error of 2.8 per cent. State the interval and what it means.
Figure (svg): A number line with the interval from 35.2 to 40.8 shaded around the estimate of 38
Subtract the margin: 38 minus 2.8 is 35.2 per cent.
Why: The lower end of the interval.
Add the margin: 38 plus 2.8 is 40.8 per cent.
Why: The upper end. The interval is centred on the estimate.
State the meaning: the true level of support among all residents plausibly lies between 35.2 and 40.8 per cent.
Why: The interval describes the population value, not the sample.
The width check catches the commonest arithmetic slip, which is adding the margin on one side only.
Verify: check the width: 40.8 minus 35.2 is 5.6, which is twice the margin of 2.8.
Why: An interval always has a full width of twice the margin, so this confirms both endpoints.
Answer: 35.2 to 40.8 per cent
Worked example
The same poll is repeated with 4,800 residents instead of 1,200. What happens to the margin of error, and roughly by how much?
Figure (svg): Bars showing the interval width halving as the sample quadruples
The sample has been multiplied by 4.
Why: 4,800 divided by 1,200 is 4.
A larger sample gives a narrower margin, so the margin shrinks.
Why: More information about the population means a more precise estimate.
Roughly, quadrupling the sample halves the margin, so it falls to about 1.4 per cent.
Why: The relationship involves the square root of the sample size, so a factor of 4 gives a factor of 2.
The SAT almost always asks only for the DIRECTION. The halving rule is a bonus, and it explains why polls rarely sample more than a couple of thousand people — the returns diminish quickly.
Verify: check the direction against intuition: asking four times as many people should give more confidence, not less.
Why: A narrower interval is exactly what more confidence means.
Answer: the margin roughly halves, to about 1.4 per cent
Worked example
Researchers randomly selected 250 members from a gym's membership list and found that 62 per cent exercise at least three times a week. To whom can this be generalised?
Figure (svg): A table showing that only the sampled population supports generalisation
Identify the sampling frame: the gym's membership list.
Why: This is the population the sample was drawn from.
The selection was random within that list, which licenses generalisation to it.
Why: Random selection makes the sample representative of the frame it was drawn from.
So the result applies to members of this gym, and to no wider group.
Why: Nothing in the study speaks to people outside the membership list.
The gym example makes the bias visible. On the test the sampling frame is often less obviously unrepresentative, but the rule is identical.
Verify: check the obvious bias against a wider claim: gym members exercise more than the general population by definition.
Why: Generalising to all adults would be badly wrong, which confirms the restriction matters.
Answer: only to the members of that gym
Step zero
Before any arithmetic.
Discussion prompt
A question describes a survey and asks which conclusion is supported. What do you locate in the stem before reading any of the choices?
Hint: One sentence decides most conclusion questions.
Answer:
Find the sentence describing who was sampled, and how they were chosen.
That sentence decides the scope of every conclusion. The result applies to the population the sample was drawn from, and to no wider group.
Also check HOW they were chosen. Randomly selected licenses generalisation to that frame; self-selected or volunteers licenses nothing at all.
Then read the choices and reject every one naming a group broader than the sampling frame.
Why this comes first: it eliminates most of the choices before you have looked at a single number, and the numbers are rarely what the question turns on.
Worked example
A website asks visitors to click a link to complete a survey about internet usage. Of the 5,000 who responded, 89 per cent said they use the internet daily. What is wrong?
Figure (svg): Two dot plots showing the self-selected respondents clustered far above the true population
Identify how the sample was chosen: visitors chose to respond themselves.
Why: This is a self-selected or voluntary-response sample, not a random one.
The sample is biased: people who visit a website and click a survey link are heavy internet users.
Why: The selection method systematically favours one kind of respondent.
The large sample size does not fix this. It makes the estimate precise about the wrong population.
Why: Sample size affects precision, not bias.
This is the crucial point about bias: more data does not help. Only a change in how the sample is drawn would.
Verify: ask what population 89 per cent actually describes: engaged visitors to that website.
Why: That group genuinely does use the internet daily, so the number is not wrong — the generalisation is.
Answer: the sample is self-selected and biased, so the result does not generalise to any wider population
Worked example
Method A has a success rate estimated at 71 per cent with a margin of 5, and Method B at 76 per cent with a margin of 5. Does the study show that B is better?
Figure (svg): A number line showing the two intervals overlapping between 71 and 76
Build both intervals: A runs from 66 to 76, and B runs from 71 to 81.
Why: Each is the estimate plus or minus its margin.
The intervals overlap between 71 and 76.
Why: Any value in that band is plausible for both methods.
So the true rates could be equal, and the study has not established that B is better.
Why: Overlapping intervals mean the difference has not been demonstrated.
Note the careful wording. The study does not show B is better; it also does not show the methods are equal. Failing to establish a difference is not evidence of sameness.
Verify: check what would settle it: non-overlapping intervals, which a larger sample might produce by narrowing both.
Why: The right response is more data, not a stronger claim from this data.
Answer: no — the intervals overlap, so a real difference has not been established
Faded example
From memory. Four lines that carry the type.
Fill in the blanks
A confidence interval is the estimate plus or minus the margin of error. A larger sample makes the interval narrower. The margin depends on the size of the sample, not of the population. And results generalise only to the population that was sampled.
Why: The third blank is the one that surprises people. A poll of 1,000 is about as precise for a country as for a town, which is why national polls do not need enormous samples. What matters is how many you asked and whether they were selected properly.
Worked example
A study estimates the mean daily screen time of a town's teenagers at 4.2 hours, with a margin of error of 0.3 hours. Is it correct to say most teenagers spend between 3.9 and 4.5 hours on screens?
Figure (svg): Two dot plots contrasting the narrow interval for the mean with the wide spread of individuals
The interval 3.9 to 4.5 is a plausible range for the MEAN screen time of all teenagers in the town.
Why: A confidence interval describes a population average or proportion.
It says nothing about how individual teenagers are distributed.
Why: Individual values vary far more widely than an average does.
So the statement is wrong: many teenagers will fall well outside that range.
Why: Some spend under an hour and some spend eight hours, while the average sits at 4.2.
This confusion between the precision of an average and the spread of the data is a distractor the SAT builds deliberately, and it is worth recognising on sight.
Verify: check with the plotted individuals: several fall far outside 3.9 to 4.5, yet their mean is inside it.
Why: A narrow interval for a mean is entirely compatible with a wide spread of individuals.
Answer: no — the interval describes the mean, not the range of individual values
Fill the middle
The word error is misleading.
Fill in the blanks
A margin of error measures uncertainty caused by sampling rather than by mistakes. It does not fix a biased sample. It describes a plausible range for a population average, not for individual values. And two intervals that overlap do not establish a difference.
Why: All four blanks are things students assume a margin of error does and it does not. The most consequential is the second: bias is a property of how the sample was chosen, and no amount of extra data corrects it.
Estimation
A quick check on the arithmetic.
Predict first
An estimate of 61 per cent has a margin of error of 4 per cent. How wide is the whole interval?
Correct: 8 percentage points
Why: The interval runs from 57 to 65, so its full width is 8 — twice the margin. The margin measures the distance from the centre to either end, not the total width. This distinction is worth carrying because it is the fastest check on an interval calculation: if the endpoints are not exactly twice the margin apart, one of them is wrong.
Check
Find who was sampled.
Check your understanding
A researcher randomly selected 500 subscribers from a magazine's subscriber list and found 31 per cent read it weekly. Which conclusion is best supported?
Answer: A
Why: The sample was drawn randomly from that magazine's subscriber list, so the result generalises to that list and no further. The word about correctly acknowledges that a sample estimate carries uncertainty.
Choices B and C widen the scope and D over-claims precision. Those are the two failure modes to scan for on every conclusion question.
Section
Section 4
Trap
The trap. A survey of 400 students at one university finds 68 per cent own a car, and a choice concludes that 68 per cent of all university students own cars.
The sample was drawn from one university. Nothing about it speaks to any other institution, whose students may differ in income, location and transport options.
The random selection was WITHIN that university, so it licenses conclusions about that university only.
The fix. Locate the sampling frame in the stem and treat it as a boundary.
Reject any choice naming a group wider than the one actually sampled.
Trap
The trap. A voluntary online survey gets 50,000 responses, and a choice argues the result is reliable because the sample is huge.
Sample size affects PRECISION, not BIAS. A self-selected sample of 50,000 estimates the views of people who choose to respond, very precisely.
Increasing it further just sharpens an estimate of the wrong population.
The fix. Check HOW the sample was chosen before considering how large it is.
Random selection prevents bias; size only narrows the interval.
Error analysis
A student's conclusion from a survey. The arithmetic is right and the scope is not.
Annotate
On: \( \text{400 students at Lincoln High, 62 percent walk} \;\Rightarrow\; \text{62 percent of teenagers walk to school} \)
The scope of a conclusion is set by the sampling frame, not by the size of the sample or the quality of the arithmetic. Find the frame first.
Trap
The trap. An interval of 3.9 to 4.5 hours for mean screen time, and a choice says most teenagers spend between 3.9 and 4.5 hours on screens.
The interval describes the plausible range for the AVERAGE, which is a single number about the whole population.
Individuals vary far more widely, and many will fall well outside it.
The fix. Ask what quantity the interval is about: a mean, a proportion, or an individual?
It is never about an individual.
Elimination
A city randomly surveyed 600 of its own residents; 44 per cent supported a transport levy, with a margin of error of 4 per cent.
Eliminate the wrong options
Which conclusion is supported? Three can be eliminated on scope or precision.
Survives elimination: a
Why: The interval is 44 plus or minus 4, giving 40 to 48 per cent, and the population is the city's residents because that is who was sampled. Each wrong choice fails in a different way: one is too precise, one is too broad, and one draws a stronger conclusion than the interval supports. Those three failure modes cover nearly every distractor on this type.
Trap
The trap. Two groups estimated at 52 and 55 per cent, both with a margin of 4, and a choice concludes the second group is higher.
The intervals are 48 to 56 and 51 to 59, which overlap. The true values could be identical.
The point estimates do differ, but the survey has not established that the populations do.
The fix. Build both intervals and check whether they overlap.
Overlapping means no difference has been established; non-overlapping suggests a real one.
Counterexample
A claim about sample size that sounds sensible.
Discussion prompt
A student says: a survey of 1,000 people is enough for a town but not for a country, because a country has far more people. Explain why this is wrong.
Hint: What does the margin of error actually depend on?
Answer:
The margin of error depends on the SAMPLE size, not the population size. A properly drawn sample of 1,000 gives about the same precision either way.
An analogy that makes it intuitive: to judge whether a pot of soup is salty enough, you taste a spoonful. It does not matter whether the pot holds two litres or two hundred — provided the soup is stirred, one spoonful tells you as much.
Stirring is the random selection. If the soup is not stirred, no amount of tasting from the top helps, which is the bias point.
This is why national polls use samples of about a thousand. Going to ten thousand costs ten times as much and only narrows the margin by a factor of about three.
The one caveat: the sample must be a small fraction of the population, which it essentially always is in these questions.
Edge cases
Results extend to the population sampled. What exactly is that population?
Discussion prompt
How do you identify the sampling frame precisely, and what makes a sample fail to represent even the group it was drawn from?
Hint: Two different things can go wrong.
Answer:
The sampling frame is the list or group the sample was actually drawn from — a membership list, a school roll, a customer database, a phone directory.
Conclusions extend to that frame and no further, however similar a wider group may seem.
Even within the frame, a sample can fail if selection was not random: volunteers, convenience samples, or people who happened to be in one place at one time.
A second failure within the frame is non-response. If only 10 per cent of those contacted reply, the respondents may differ systematically from the rest, and the effective sample is self-selected again.
So there are two questions: was the frame the population you want, and was the selection within it random? Both must be yes.
On the SAT the first question is asked far more often, and the correct answer usually restricts the conclusion to exactly the group named in the stem.
Check
How the sample was chosen matters more than how big it is.
Check your understanding
A radio station invites listeners to phone in and vote on a proposal. Of 20,000 callers, 78 per cent were in favour. What is the best assessment?
Answer: A
Why: Callers chose to participate, so the sample consists of people motivated enough to phone in — who are likely to hold stronger views than average. Size does not correct this: a large biased sample gives a precise estimate of a group that was never the target population.
Choice C is the most instructive distractor: it is arithmetically true and completely misleading, which is precisely why bias matters more than size.
Section
Section 5
Matching
Six pieces of vocabulary, six definitions.
Match the pairs
Why: This is genuinely the whole type. Six definitions, one addition and one subtraction, and two rules about scope. An hour spent on this card is worth more than an hour of practice questions, which is why a 1.3 per cent type earns its own deck.
Sorting
A random sample of 400 residents of one town was surveyed.
Sort into buckets
Is the conclusion supported?
The three unsupported conclusions are the three standard distractor types: too broad, too precise, and about individuals rather than the average.
Comparison
Fill the blanks from memory. Two different problems with two different fixes.
Comparison matrix
| precision | bias | |
|---|---|---|
| What it is | how narrow the interval is | systematic misrepresentation of the population |
| Caused by | sampling only part of the population | a selection method that favours some members |
| Fixed by | a larger sample | random selection, never by size |
| Measured by | the margin of error | nothing — it is not quantified at all |
The bottom right cell is the important one. A margin of error quantifies precision and says nothing whatever about bias, which is why a biased survey can report a very small margin and still be worthless.
Trade off
Fill in what each change actually achieves.
Comparison matrix
| change to the study | what improves | what does not |
|---|---|---|
| Survey four times as many people | the margin roughly halves | bias is unaffected |
| Select respondents randomly | the sample represents the frame | the interval width is unchanged |
| Broaden the sampling frame | conclusions apply to a wider group | precision is unchanged |
| Report a smaller margin | nothing at all | it is a consequence, not a choice |
The bottom row matters: a margin of error is computed from the study, not selected by the researcher. A study cannot become better by reporting a narrower margin.
Real world
One minute on why polls disagree.
Discussion prompt
Two polls published in the same week report 46 per cent and 51 per cent support for the same policy. Do they contradict each other?
Answer:
Probably not. If each has a margin of about 3, the intervals are 43 to 49 and 48 to 54, which overlap between 48 and 49.
So both are consistent with a true value near 48.5 per cent, and neither contradicts the other.
This is why reporting a single headline number is misleading. The margin is usually printed in small type at the bottom, and it is what makes the two figures compatible.
It also explains the phrase within the margin of error, used when a lead is too small to be meaningful. A candidate ahead by 2 points with a margin of 3 is not established as ahead at all.
And it explains why polls can be precise and wrong. A large but biased sample — one that under-represents a group of voters — reports a small margin around the wrong number, which is exactly the failure mode this type is teaching you to spot.
Ranking
All four surveys measure the same quantity in the same population. Order from smallest margin to largest.
Put in order
Why: The margin shrinks as the sample grows, so the order from smallest margin to largest is 1,600, then 400, then 100, then 25. The relationship involves the square root: quadrupling the sample halves the margin, so if the sample of 100 gives a margin of 10 points, then 400 gives about 5 and 1,600 gives about 2.5, while 25 gives about 20. Note that the population size never entered the reasoning.
Concept
This type is 1.3 per cent of the section — about one question — and it is almost pure vocabulary, which makes it the best return per hour on the whole Math section.
| session | what you do | why |
|---|---|---|
| 1 | Learn the six terms cold: population, sample, statistic, parameter, margin, interval. | The whole type is built on this vocabulary, and questions are unanswerable without it. |
| 2 | Build fifteen intervals, checking each is exactly twice the margin wide. | The only arithmetic in the type, and the width check catches every slip. |
| 3 | Ten conclusion questions, underlining the sampling frame before reading any choice. | Scope is what most of these questions actually test. |
| 4 | Ten questions mixing sample size with bias, deciding which issue each one raises. | Separating precision from bias is the distinction the hardest questions turn on. |
Naruhodo Tutoring — SAT Math question bank export (sat-question-index.json) — filter the bank to this skill tag — 22 questions, roughly a third of each difficulty
Explain it to yourself
Close the deck.
Discussion prompt
Without looking, state what a larger sample does, what it does not do, and how far a survey result can be generalised.
Hint: One rule about precision, one about scope.
Answer:
A larger sample narrows the confidence interval, because more data gives a more precise estimate of the population value.
It does not fix bias. If the sample was chosen in a way that favours some members, a larger sample estimates the wrong population more precisely.
The margin depends on the sample size, not the population size — a sample of 1,000 is about as precise for a country as for a town.
Results generalise only to the population that was sampled, which is the frame the sample was drawn from.
Random selection is what licenses that generalisation, and it licenses nothing beyond the frame.
Explain it
Two minutes, out loud.
Discussion prompt
A friend says a survey with 50,000 responses must be more reliable than one with 800. When are they wrong?
Answer:
Grant the general case: with the same sampling method, more is genuinely better and the interval narrows.
Then give the exception: if the 50,000 chose to respond themselves and the 800 were randomly selected, the smaller survey is far more trustworthy.
Make it concrete. A website poll with 50,000 self-selected responses measures the views of people who visit that site and feel strongly. That is a real population — just not the one anyone wanted to know about.
Give the soup analogy: tasting a spoonful tells you about the pot only if the soup was stirred. Random selection is the stirring; sample size is how big the spoon is. A huge spoon from an unstirred pot still tells you about the top layer.
Then the rule: ask HOW the sample was chosen before asking how big it is. Size buys precision; only randomness buys representativeness.
Commit first
Commit before you check.
Predict first
A study randomly sampled 800 patients at one hospital and found a treatment effective for 73 per cent. The result generalises to
Correct: patients at that hospital
Why: The sample was drawn randomly from one hospital's patients, so it represents that hospital's patients and no wider group. Patients elsewhere may differ in age, severity, or the presence of other conditions, and nothing in the study speaks to them. Each broader choice claims more than the sampling frame supports — which is the single most common error on this type.
Connect it up
Blank paper.
Draw it
Draw a large circle labelled POPULATION and a small circle inside it labelled SAMPLE, with an arrow from the population to the sample marked random selection. From the sample, draw an arrow out to a number line showing an estimate with a margin either side, labelled CONFIDENCE INTERVAL, and note that its full width is twice the margin. Beside the diagram write the two rules: a larger sample narrows the interval, and conclusions extend only to the population sampled. In a separate box, write what a margin of error is NOT: not about mistakes, not a fix for bias, and not a range for individuals. At the bottom, write the soup analogy in one line: stir the pot, then one spoonful is enough however big the pot is.
Exit ticket
One question before you close the deck.
Predict first
A survey has a very large sample but respondents volunteered themselves. What is the main problem?
Correct: The sample is biased, and size does not correct that
Why: Volunteers differ systematically from those who do not volunteer, so the sample misrepresents the target population no matter how many respond. The margin of error will actually be very NARROW, which is what makes this dangerous — it suggests a precision the result does not deserve. An interval can be computed; it simply describes the wrong population. Size buys precision, and only random selection buys representativeness.
Recap
One type, almost pure vocabulary: know six terms and two rules.
| never do this | do this instead |
|---|---|
| Generalise beyond the group actually sampled | Restrict the conclusion to the sampling frame |
| Trust a huge self-selected sample | Check how the sample was chosen before how big it is |
| Say exactly, when a margin is given | Say about, or give the interval |
| Read the interval as a range for individuals | It is a range for the population average |
| Claim a difference from overlapping intervals | Say the difference has not been established |
| Assume a bigger population needs a bigger sample | The margin depends on the sample alone |
Naruhodo Tutoring — SAT Math question bank export (sat-question-index.json) — and the site's SAT pages to drill this type in isolation, then mixed
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