Inference From Sample Statistics and Margin of Error Pattern - Estimation

Inference: Estimation

This pattern tests one core skill: using a sample proportion to estimate a count in a larger population. A random sample reveals that some percentage (or fraction) of items have a certain property. You then apply that proportion to the total population to estimate how many items in the whole group share that property.

The Formula

$\text{Estimated count} = \text{population size} \times \text{sample proportion}$

The sample proportion can be given as a percentage (convert to a decimal first) or as a fraction.

Two Formats You Will See

1. Percentage given directly. "A random sample found that 4% of 1,500 smartphones had a defect." → $1500 \times 0.04 = 60$

2. Fraction computed from sample counts. "Out of 50 fish caught, 45 were trout. Estimate the trout in a lake of 1,200 fish." → Proportion $= \dfrac{45}{50} = 0.9$, then $1200 \times 0.9 = 1080$

Step-by-Step Method

1. Identify the sample proportion. If given a percentage, convert to a decimal by dividing by 100. If given sample counts, divide the count of interest by the sample size.
2. Multiply the proportion by the total population.
3. Pick the answer choice that matches.

Common Gotchas

• Confusing the percentage with the count. If 4% of items are defective, the answer is not 4 — it is 4% of the total. The raw percentage number almost always appears as a wrong answer.
• Computing the complement. If 15% are infected, then 85% are not infected. The SAT puts $\text{population} \times 0.85$ as a trap answer. Make sure you are estimating the group the question asks about.
• Subtracting instead of multiplying. Some wrong answers are $\text{population} - \text{sample size}$ or $\text{population} - \text{count in sample}$. These operations make no statistical sense — always multiply.

Worked Example 1

A factory produced 1,500 smartphones. A random sample found that 4% had a screen defect. Estimate the total number of defective phones.

A) 4 $\quad$ B) 60 $\quad$ C) 1,440 $\quad$ D) 1,500

SOLUTION

Sample proportion $= 4\% = 0.04$
Estimated count $= 1500 \times 0.04 = 60$
Answer: B) 60

A) 4 is the percentage, not the count.
C) 1,440 is the number of phones without defects ($1500 \times 0.96$).
D) 1,500 is the total population.

Worked Example 2

A biologist catches 50 fish from a lake of 1,200 fish. Of the sample, 45 are trout. Estimate the total trout in the lake.

A) 5 $\quad$ B) 1,080 $\quad$ C) 1,150 $\quad$ D) 1,155

SOLUTION

Sample proportion $= \dfrac{45}{50} = 0.9$
Estimated count $= 1200 \times 0.9 = 1080$
Answer: B) 1,080

A) 5 is the number of non-trout in the sample ($50 - 45$).
C) 1,150 is $1200 - 50$ (subtracting the sample size — meaningless).
D) 1,155 is $1200 - 45$ (subtracting the trout count — also meaningless).

What to Do on Test Day

• Formula: $\text{Estimated count} = \text{population} \times \text{sample proportion}$
• Convert percentages to decimals before multiplying.
• If given raw counts in a sample, compute $\dfrac{\text{count of interest}}{\text{sample size}}$ first.
• The percentage itself, the complement count, and any subtraction-based number are always traps.
• These are fast questions — 20 to 30 seconds if you stay focused on the formula.