Scale a Sample Proportion into an Estimate

Supplied materials

• Paper or a browser scratch area; a calculator is optional unless rounding is requested.
• Supplied teaching case: In a random sample of 40 from a fictional population of 400, 20 favor a proposal. Use the sample proportion to estimate the population number favoring it. Candidate answers: (A) 199 people (B) 200 people (C) 201 people (D) 400 people.
• Independent check inputs: In a random sample of 40 from a fictional population of 400, 21 favor a proposal. Use the sample proportion to estimate the population number favoring it. Candidate answers: (A) 209 people (B) 210 people (C) 211 people (D) 420 people.

Learn the idea

A complete population, a selected sample and a probability model are different sources of information. Sampling supports estimates, observed trial counts support experimental proportions, and explicit independence permits multiplication of model probabilities. A sample proportion divides the counted category by the sample size. Applying that proportion to population size creates an estimated population count, with uncertainty from sampling variation.

Here is the method on the supplied teaching case. The sample proportion is 20/40. Multiply by 400: (20/40)×400=200. This is an estimate with sampling variation, not an exact count.

An estimate is not a census total. State the random-sampling assumption and the observed sample denominator; repeating the sampling process could give a different proportion and therefore a different estimate.

1. Write or sketch the supplied teaching case, labeling its quantities and restrictions.
2. Apply the method shown in the worked solution, keeping each intermediate equality or justified inference.
3. Solve the independent check with its changed inputs before revealing feedback.
4. Complete the three practice cases; explain which mathematical rule rules out one distractor in each case.

Check your reasoning

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Take it further

Changed-input task: In a random sample of 40 from a fictional population of 400, 27 favor a proposal. Use the sample proportion to estimate the population number favoring it. Candidate answers: (A) 269 people (B) 270 people (C) 271 people (D) 540 people. Show the mathematical reason, not only the selected answer. Expected result: 270 people. The sample proportion is 27/40. Multiply by 400: (27/40)×400=270. This is an estimate with sampling variation, not an exact count.

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