Estimating incorrect tax returns using a sample proportion

Key takeaway: This question uses proportional reasoning: a small audited sample is used to estimate a large population total.

Set up the proportion

The problem says 10,000 tax returns were audited and 500 were incorrect. That means the error rate in the sample is:

$\frac{500}{10000} = 0.05$

So about 5% of the returns were incorrect. To estimate the number of incorrect returns out of 250,000,000 filed returns, multiply the total by the same proportion:

$250{,}000{,}000 \times 0.05 = 12{,}500{,}000$

So the best estimate is 12,500,000, which is choice (b).

Why proportional reasoning works here

This is a sampling problem. The audited returns form a representative sample, and the incorrect rate in that sample is used to estimate the same rate in the full population. The reasoning is not exact counting; it is a prediction based on proportion.

A faster way to see it is to convert the sample rate into a fraction:

$\frac{500}{10000} = \frac{1}{20}$

That means about 1 out of every 20 returns is incorrect. Applying that to 250,000,000 gives:

$250{,}000{,}000 \div 20 = 12{,}500{,}000$

Check the answer against the choices

The answer must be much larger than 12,500 because the full population is 25,000 times larger than the sample. It must also be much smaller than 125,000,000 because only 5% are incorrect, not 50%.

This kind of question tests whether you can move from a sample to a population using the same ratio. The central skill is recognizing percentage rate, not doing complicated arithmetic.

Common reasoning pattern

When a sample is described with both a total and a number of failures, first compute the fraction or percentage. Then apply that rate to the larger total. If the sample is assumed to represent the whole group, the estimate should scale directly with the population size.

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Pitfalls the pros know 👇 Students often choose 500 because they focus only on the audited sample and forget that the question asks about all 250,000,000 returns. Another error is multiplying by the wrong ratio, such as using 10,000/500 instead of 500/10,000. That flips the meaning of the percentage and gives a wildly incorrect total. Always compute the rate of incorrect returns first, then scale up.

What if the problem changes? If the problem changed to 20,000 audited returns with 800 incorrect, the incorrect rate would still be 4%, and the estimate for 250,000,000 returns would be 10,000,000. If the question instead asked for the number of correct returns, you would subtract the estimated incorrect number from the total. For example, with 12,500,000 incorrect returns, the estimated correct returns would be 237,500,000.

Tags: sample proportion, population estimate, percentage rate