Inference From Sample Statistics and Margin of Error Pattern - Estimate With Error

Inference: Estimate with Error

This pattern combines the ideas from the previous two inference patterns. You are given a sample proportion and a margin of error, and you must determine a plausible count for the full population. Instead of just multiplying one percentage by the population, you first build a range of plausible percentages, then scale that range up to the population level, and finally pick the answer choice that falls within that range.

The Method

1. Build the percentage interval: $[\text{estimate} - \text{MOE},\ \text{estimate} + \text{MOE}]$.
2. Convert both bounds to decimals.
3. Multiply each bound by the population size to get a numerical range.
4. The correct answer is the choice that falls within this range.

Step-by-Step Example

A university has 20,000 students. A survey estimates 24% are STEM majors with a margin of error of 4%.

• Percentage range: $24\% \pm 4\% = [20\%,\ 28\%]$
• Lower bound: $20{,}000 \times 0.20 = 4{,}000$
• Upper bound: $20{,}000 \times 0.28 = 5{,}600$
• Plausible count: any value between 4,000 and 5,600

Common Gotchas

• Applying the proportion to the sample instead of the population. If the sample has 500 students and 24% are STEM, then $0.24 \times 500 = 120$ — but this is the count in the sample, not the population. This is always a wrong answer.
• Computing the complement count from the sample. $500 - 120 = 380$ — the number of non-STEM students in the sample. Also a wrong answer.
• Giving the complement of the population. If 24% are STEM, then 76% are not. The SAT sometimes places a value in the 72%–80% range (the complement's interval) as a wrong answer.

Worked Example 1

A factory produced 80,000 smartphones. An inspection of 2,000 phones estimates 6% have a cosmetic defect, with a margin of error of 1.5%. Which could be the actual number of defective phones?

A) 120 $\quad$ B) 5,200 $\quad$ C) 1,880 $\quad$ D) 75,000

SOLUTION

Percentage range: $6\% \pm 1.5\% = [4.5\%,\ 7.5\%]$
Lower bound: $80{,}000 \times 0.045 = 3{,}600$
Upper bound: $80{,}000 \times 0.075 = 6{,}000$
Plausible range: 3,600 to 6,000

A) 120 $= 0.06 \times 2{,}000$ → defective in the sample, not the batch.
C) 1,880 $= 2{,}000 - 120$ → non-defective in the sample.
D) 75,000 is plausible for the non-defective population count.
B) 5,200 is between 3,600 and 6,000 ✓
Answer: B) 5,200

Worked Example 2

Biologists estimate 15% of 4,000 birds in a refuge are juvenile, with a margin of error of 5%. Which is a plausible number of juvenile birds?

A) 60 $\quad$ B) 650 $\quad$ C) 3,300 $\quad$ D) 340

SOLUTION

Percentage range: $15\% \pm 5\% = [10\%,\ 20\%]$
Lower bound: $4{,}000 \times 0.10 = 400$
Upper bound: $4{,}000 \times 0.20 = 800$
Plausible range: 400 to 800

A) 60 $= 0.15 \times 400$ → juvenile count in the sample.
D) 340 $= 400 - 60$ → adult count in the sample.
C) 3,300 is in the range for adult birds in the population.
B) 650 is between 400 and 800 ✓
Answer: B) 650

What to Do on Test Day

• Build the percentage interval first: $\text{estimate} \pm \text{margin of error}$.
• Scale both endpoints to the population (not the sample).
• The correct answer falls between the lower and upper bounds.
• Three classic traps: (1) the count in the sample, (2) the complement count in the sample, (3) the complement count in the population. Recognize them and move on.
• Write down the numerical range before scanning the answer choices — it keeps you from being lured by trap answers.