NavGanitMateby NavGreen Compute, AustraliaBased on Victorian Curriculum Mathematics v2.0
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Sample size and variability

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Two random samples of 32 and 24 students from a school in Melbourne are taken to estimate the proportion who prefer art class over science class. In the first sample, 18 students prefer art class.

In the second sample, 12 students prefer art class. What is the exact difference between the proportion of students who prefer art class in the two samples?

Express your answer as a simplified fraction.

Concept illustrations

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Larger samples can vary less. Illustrative proportions from repeated samples: n=10 gives 0.2,0.4,0.5,0.6,0.8; n=100 gives 0.45,0.48,0.50,0.52,0.55. These are authored illustrations, not measured experiment results or a guarantee.n=10n=10000.51Observed sample proportion
Illustrative proportions from repeated samples: n=10 gives 0.2,0.4,0.5,0.6,0.8; n=100 gives 0.45,0.48,0.50,0.52,0.55. These are authored illustrations, not measured experiment results or a guarantee.
A sample is part of a population. The large outline represents a population of 24 members. Six filled members illustrate a sample. This picture does not establish that a selection process was random.6 selected from 24
The large outline represents a population of 24 members. Six filled members illustrate a sample. This picture does not establish that a selection process was random.

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Explanation

Worked solution and answer (click to reveal)

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Worked solution

  1. The first sample has 32 students, with 18 preferring art class. The proportion is calculated as 18 divided by 32.
  2. Compute the proportion: 18 / 32 = 0.5625. This is the decimal form of the proportion.
  3. The second sample has 24 students, with 12 preferring art class. The proportion is 12 divided by 24.
  4. Compute the proportion: 12 / 24 = 0.5. This is the decimal form of the second proportion.
  5. Find the difference between the two proportions: 0.5625 - 0.5 = 0.0625.
  6. Convert 0.0625 to a fraction. Since 0.0625 = 625 / 10000, simplify by dividing numerator and denominator by 125: 625 ÷ 125 = 5, 10000 ÷ 125 = 80.
  7. Simplify 5 / 80 by dividing both by 5: 5 ÷ 5 = 1, 80 ÷ 5 = 16.
  8. The simplified fraction is 1 / 16.
  9. Verify the fraction: 1 / 16 = 0.0625, which matches the decimal difference.
  10. The exact difference between the proportions is 1 / 16.

Reference answer: 1/16

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