Inference from Sample Statistics, Margin of Error

Inference means using a sample to say something about a whole population. The SAT asks you to scale a sample result up to a population estimate, to read and compare margins of error and confidence intervals, and to judge which sampling method gives trustworthy results. Most questions need careful reading more than calculation.

• A random sample represents the population it was drawn from. That is what lets you scale its results up.
• To estimate a count in the population, multiply the sample proportion by the population size.
• A sample result is an estimate. The margin of error gives a range of plausible values: estimate $\pm$ margin of error.
• Larger random samples give smaller margins of error. Sample size does not fix bias, though. A large biased sample is still biased.
• A confidence interval is about a population parameter (such as the mean or the percent), not about individual members.
• If two intervals overlap, the data do not clearly show which population value is larger. If they do not overlap, there is convincing evidence of a difference.

$\text{population estimate} = \frac{\text{count in sample}}{\text{sample size}} \times \text{population size}$

• To estimate how many do not have a trait, use the complement: $1 - \text{sample proportion}$.

An inspector tests a random sample of $250$ phones from a shipment of $15{,}000$ and finds $9$ defective. Estimate the number of defective phones in the shipment.

1. Sample proportion: $\frac{9}{250} = 0.036$.
2. Scale up: $0.036 \times 15{,}000 = 540$.

Check with a proportion: $\frac{9}{250} = \frac{x}{15{,}000}$ gives $x = 9 \times 60 = 540$. Correct.

In a random survey of households in Town A, the plausible values for the percent that own an electric bike are from 41.2% to 46.8%. In Town B, a similar survey estimated 49% with a margin of error of $2$ percentage points. What was Town A's estimate and margin of error, and what can be concluded?

1. Town A's estimate is the midpoint: $\frac{41.2 + 46.8}{2} = 44$, so 44%.
2. Its margin of error is half the width: $\frac{46.8 - 41.2}{2} = 2.8$ percentage points.
3. Town B's interval runs from 47% to 51%.
4. The intervals do not overlap (A tops out at 46.8%), so there is convincing evidence that a greater percent of households in Town B own an electric bike.

If Town B's margin had been $3$ points, its interval would start at 46%, overlapping Town A's, and no clear conclusion could be drawn.

• Applying an interval to individuals. "The mean nightly sleep is plausibly between $6.8$ and $7.4$ hours" does not mean most students sleep that long.
• Thinking a bigger sample removes bias. A survey of $5{,}000$ volunteers can still misrepresent the population.
• Scaling to the wrong population. A sample of seniors at one school tells you about seniors at that school, not about all students.
• Forgetting the complement. If the question asks how many are not infected, subtract from the total or use $1 - p$.
• Doubling the margin incorrectly. The margin of error is half the interval's width, not the full width.