SAT Math Review · Problem-Solving and Data Analysis
Sampling, margin of error and inference
A sample never gives you the exact truth about a population. It gives you a range, and a question about which sentences that range will support.
Which conclusion is best supported?
A restaurant chain selected 250 of its customers at random and recorded the amount each spent per visit. Based on the sample, the mean amount spent per visit by all of the chain's customers is estimated to be $34.80, with an associated margin of error of $2.40.
The margin of error surrounds the estimate with an interval of plausible values for the population mean: 34.80 − 2.40 = 32.40 and 34.80 + 2.40 = 37.20, so C is the only supported sentence. A drops the margin of error and treats a sample estimate as exact; if it were exact there would be no need to report a margin at all. B and D apply the interval to individual customers, which it never describes — plenty of single customers spend $9 or $60. The interval is about the average of everybody, not about anybody.
1
The sample is measured; the population is estimated.
Nobody surveys all 12,000 customers. A manageable group is measured, and its result is used as the best guess for the group nobody measured.
population = everyone the question is aboutsample = the few actually measuredestimate = the sample's result, reused
Scaling a percent up to a count is one of the two calculations in this topic, and it uses the population size, never the sample size.
400 of a town's 9,000 registered voters, chosen at random
sample 400, population 9,000
232 of the 400 support the measure
232 ÷ 400 = 0.58, a measured fact about the sample
0.58 × 9,000 = 5,220
the estimate for the whole town
The wrong multiplication
0.58 × 400 = 232 is just the count you started with, and it will be offered as a choice. The question asks about the town, so the percent has to be applied to the town's 9,000.
2
The margin of error turns one number into an interval.
estimate − margin to estimate + margin. The interval is a set of plausible values for the population's true value, and the estimate sits exactly in the middle.
34.80 − 2.40 = 32.40
the low end
34.80 + 2.40 = 37.20
the high end
width = 2 × 2.40 = 4.80
the margin is the half-width, not the width
Running it backwards
Given an interval instead, the estimate is its midpoint and the margin of error is the distance from the midpoint to either end. From 32.40 to 37.20: the midpoint is (32.40 + 37.20) ÷ 2 = 34.80, and 37.20 − 34.80 = 2.40.
3
The interval describes the population's average, and nothing else.
Three of the four choices on an SAT inference item are usually true statements about the wrong thing, or overstated statements about the right thing.
Sentence
Supported?
Why
It is plausible that the population mean is between 32.40 and 37.20.
yes
exactly what the interval says
The population mean is exactly 34.80.
no
the margin of error exists because it is not exact
Every customer spent between 32.40 and 37.20.
no
about individuals, not about the mean
It is impossible that the mean is 38.00.
no
values outside are unlikely, not impossible
The mean for the 250 sampled customers is 34.80.
yes
that is the measurement the estimate came from
"Plausible", "it is likely that", and "based on this estimate" are the words of a correct choice. "Exactly", "every", "all", "must", "impossible" and "proves" almost always mark a wrong one.
4
Random selection is what lets the estimate leave the sample.
A margin of error is calculated as if the sample were drawn at random from the population being described. If it was not, no arithmetic repairs it.
250 customers chosen at random from the chain's list
generalises to all of the chain's customers
the population the sample was drawn from
250 customers who filled in a card at one branch
generalises to nobody
volunteers at one location are not the chain
The sample also fixes which population is being described. A random sample of a club's members supports a conclusion about that club's members — not about everyone in the country who shares the hobby.
A big sample does not fix a bad one
Surveying 5,000 shoppers outside one store is still 5,000 shoppers outside one store. The margin of error would look impressively small and would be describing the wrong group; size and selection are separate questions.
5
A larger sample makes the margin of error smaller.
Same method, same population: more people measured means a narrower interval and a sharper estimate. The size of the population it was drawn from does not change the margin, and the percent that answered yes moves it only slightly — sample size is the lever the SAT asks you about.
n = 200 → 39% to 53%
a 14-point interval: almost no information
n = 800 → 42.5% to 49.5%
half as wide, and still centred on 46%
smaller margin ⇒ larger sample
read the implication backwards too
What the SAT expects you to say
Items compare two samples from the same population, measured the same way, and ask why one margin of error is smaller. The answer is always that that sample was larger. It is worth knowing that halving the margin takes four times the sample, but the test only asks for the direction.
6
Answer the question by eliminating the overstatements.
Work the four choices against three tests, in this order.
1. whothe population, not the sampleand not one individual
2. whatthe mean or the percentnot a single person's value
3. how stronglyplausible, likelynever exactly, all, impossible
Checked against the opening problem: A fails test 3, B and D fail tests 1 and 2, and C passes all three. You can often reach the answer without computing the interval at all — but compute it anyway, because the numbers in the wrong choices are chosen to look right.
Where the calculator earns its keep
Percent-to-count conversions are where slips happen. For an estimate of 38% ± 5% of 15,000, type 0.33 × 15,000 and 0.43 × 15,000 into Desmos and read 4,950 and 6,450; then check which choice lies between them rather than re-deriving each one.
✓
Check: is the estimate still in the middle?
midpoint(32.40 + 37.20) ÷ 2 = 34.80matches the estimate ✓
half-width37.20 − 34.80 = 2.40matches the margin ✓
scaled count0.58 × 9,000 = 5,220bigger than the sample count ✓
Remember the flow
Estimate ± margin · Population, not people · Bigger sample, tighter interval
The sample decides which population you are allowed to talk about.
Now you try
Practice problem
Which conclusion is best supported?
Of the 40,000 subscribers to a magazine, 900 were selected at random and asked whether they read the digital edition. Based on the sample, it is estimated that 62% of all of the magazine's subscribers read the digital edition, with an associated margin of error of 3%.
Plausible percents run from 62 − 3 = 59% to 62 + 3 = 65%, and the question asks for subscribers, so apply both ends to the 40,000: 0.59 × 40,000 = 23,600 and 0.65 × 40,000 = 26,000. A is 0.62 × 40,000 stated as an exact count, which throws the margin of error away. C applies the interval to the 900 who were surveyed (0.59 × 900 = 531 and 0.65 × 900 = 585) rather than to the subscribers the estimate is about. D has the effect of sample size backwards: quadrupling the sample would halve the margin of error to about 1.5%.
Hint: build the percent interval first, then multiply each end by the population — and read the choices carefully enough to notice which number each one has been multiplied by.
