SAT · Problem-Solving and Data Analysis · TOPIC LESSON
Use a representative sample to estimate a population and express the uncertainty honestly.
BY THE END, YOU CAN
BEFORE YOU START
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01 / UNDERSTAND & APPLY
The population is the full group of interest; the sample is the subset measured. A statistic, such as a sample mean or proportion, is computed from the sample. A population parameter describes the whole group. Inference uses the statistic as evidence about the parameter, while acknowledging sampling uncertainty.
A random sample from a well-defined population supports generalization to that population if the sampling and response process is sound. A large convenience sample can still be biased. Population membership must match the group actually sampled.
02 / UNDERSTAND & APPLY
The sample proportion is observed successes/sample size. To estimate a population count, multiply the sample proportion by the population size. This produces an estimate, not an exact count, because a different random sample could give a different proportion.
ORIGINAL WORKED EXAMPLE
In a random sample of 200 residents, 74 favor a proposed park. Estimate how many of the town’s 5,000 residents favor it.
Answer About 1,850 residents
03 / UNDERSTAND & APPLY
A representative sample mean estimates the population’s average, not each individual value. Multiplying the estimated average by population size can estimate a population total when the measurement applies to every member and the population size is known.
Do not multiply a mean by the sample size if the requested total is for the entire population. Conversely, do not mistake the population total for the average.
ORIGINAL WORKED EXAMPLE
A random sample of 50 trees has mean fruit yield 18 kg. Estimate the total yield of 320 comparable trees in the sampled orchard.
Answer About 5,760 kg
04 / UNDERSTAND & APPLY
A reported estimate ± margin of error gives an interval around the estimate at a stated confidence level and under the procedure’s assumptions. For a proportion expressed as 58% with margin 4 percentage points, the interval is 54% to 62%; the margin is an additive width, not 4% of 58%.
The margin addresses random sampling variation under the model. It does not automatically include nonresponse, poorly worded questions, undercoverage or measurement error. A biased procedure can produce a narrow interval around the wrong target.
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ORIGINAL WORKED EXAMPLE
A survey estimates support at 58% with margin of error 4 percentage points. What interval is reported?
Answer 54% to 62%
05 / UNDERSTAND & APPLY
A 95% confidence procedure is designed so that, over many repeated samples taken and analyzed in the same way, about 95% of the resulting intervals contain the true fixed population parameter. For a particular completed interval, the fixed parameter either is or is not inside it.
A confidence interval for a population mean does not claim that 95% of individuals lie in it. It also does not guarantee that the sample mean equals the population mean. Interpret uncertainty in the parameter being estimated.
06 / UNDERSTAND & APPLY
An estimate is the center, while the interval communicates a range of values compatible with the procedure at the stated confidence level. If an interval includes 50%, an estimate slightly above 50% alone does not establish that population support exceeds half at that confidence level.
Overlapping individual confidence intervals are not a universal formal test of whether two populations differ. A direct comparison generally needs an interval or test for their difference and an appropriate design.
ORIGINAL WORKED EXAMPLE
A poll reports 52% support with margin of error 5 percentage points. Does the stated interval support an unqualified claim that a majority of the population favors the proposal?
Answer No; the reported interval includes support below 50%
07 / UNDERSTAND & APPLY
Under comparable random-sampling conditions, the same confidence level and a population large relative to the sample, margin of error typically scales approximately as 1/√n. Quadrupling sample size roughly halves the margin; merely doubling it does not halve the margin.
This approximation assumes comparable variability and design. Sampling without replacement from a substantial fraction of a small population can require a finite-population adjustment. The essential distinction is precision versus bias: more observations reduce random error but do not fix a systematically unrepresentative method.
ORIGINAL WORKED EXAMPLE
A comparable survey of 400 randomly sampled people has margin about 6 percentage points. Approximately how large a sample would halve that margin at the same confidence level?
Answer About 1,600 people
08 / UNDERSTAND & APPLY
If a population has 8,000 members and the plausible proportion interval is 0.31 to 0.39, multiplying both endpoints gives estimated counts 2,480 to 3,120. These are interval-based estimates, not verified exact counts. If rounding is requested, delay it until after multiplication.
Calculator arithmetic can evaluate endpoints, but cannot decide whether the sample represents the target population or whether a reported margin addresses all important sources of error.
09 / UNDERSTAND & APPLY
Use the reported assumptions rather than inventing certainty.
ORIGINAL PRACTICE · 1
0.30 or 30%
The sample proportion is 36/120.
ORIGINAL PRACTICE · 2
270
Multiply 0.30 × 900, while retaining the interpretation as an estimate.
ORIGINAL PRACTICE · 3
39 to 45
Subtract and add the margin to the estimate.
ORIGINAL PRACTICE · 4
62%
Percentage-point margins are additive; they are not 2% of the estimate.
ORIGINAL PRACTICE · 5
It roughly halves it
1/√(4n) is half of 1/√n.
ORIGINAL PRACTICE · 6
No
Larger size reduces random variation but does not repair the selection mechanism.
ORIGINAL PRACTICE · 7
Not necessarily
The interval estimates a population mean, not the distribution of individuals.
ORIGINAL PRACTICE · 8
Yes
The interval is 45% to 53%, which contains 50%.
ORIGINAL PRACTICE · 9
120 to 200
The proportion interval is 0.06 to 0.10; multiply each endpoint by 2,000.
KEEP THE BIG IDEAS
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