A sample, and what it can carry
Moving between a sample proportion and a population estimate, and knowing when the move is not allowed.
Notes and data
A town of 24,000 residents surveyed 400 of them, chosen at random, about library use.
Of those surveyed, 140 had borrowed a book in the past year. Of those 140 borrowers, 35 said they had also used the library computers.
Answers in order, explanations withheld, and a report on where the chain broke.
- 8.1PercentagesFoundation
What percentage of the people surveyed had borrowed a book in the past year?
- A35%
- B25%
- C14%
- D40%
Show the answer and reasoningHide the reasoning
35%
140 out of 400 is 140 / 400 = 0.35, which is 35 percent.
- 8.2Proportional relationshipsStandard
Builds on: The 35 percent from question 1 — this step is that proportion applied to the whole town.
Roughly how many of the town's 24,000 residents would you estimate had borrowed a book?
- AAbout 8,400
- BAbout 140
- CAbout 2,100
- DAbout 6,000
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About 8,400
Applying the sample proportion to the population: 0.35 x 24,000 = 8,400. The estimate is only as good as the sample being random, which the stimulus states it was.
- 8.3Two-way frequenciesChallenge
Builds on: The distinction question 1 set up between a count and the group it is counted out of — the group changes here.
What percentage of the surveyed borrowers had also used the library computers?
- A25%
- B8.75%
- C35%
- D14%
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25%
The denominator here is borrowers, not everyone surveyed: 35 / 140 = 0.25, which is 25 percent. Dividing 35 by 400 gives 8.75 percent, which answers a question about the whole sample instead.
- 8.4SamplingChallenge
Builds on: Question 2, which established that the sample can support a population estimate, and question 3, which established whose subset the 35 is.
The library wants to estimate how many residents use its computers. What is the main limitation of doing so from this survey?
- AComputer use was only asked of borrowers, so residents who use the computers without borrowing are missing.
- BThe sample of 400 is too small to estimate anything about 24,000 people.
- CPercentages cannot be applied to a population.
- DThe survey did not record how often each person visited.
Show the answer and reasoningHide the reasoning
Computer use was only asked of borrowers, so residents who use the computers without borrowing are missing.
The 35 figure is a subset of the 140 borrowers, so it says nothing about computer users who never borrow. That is a coverage problem in the question asked, not a problem with the sample size — question 2 showed a random sample of 400 supports a population estimate perfectly well.