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

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Required skills (click to reveal)

Two random samples of walkers were taken from the same population in Darwin to compare completion rates for a 67 km walking route. Sample A, of size 54, had 31 walkers complete the route.

Sample B, of size 67, had 42 walkers complete the route. What is the exact difference between the proportion of walkers who completed the route in Sample A and Sample B?

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. Identify the given data: Sample A has 54 walkers, 31 completed the route. Sample B has 67 walkers, 42 completed the route.
  2. Calculate the proportion of completions in Sample A: 31 divided by 54.
  3. Compute the fraction: 31/54. This is the proportion for Sample A.
  4. Calculate the proportion of completions in Sample B: 42 divided by 67.
  5. Compute the fraction: 42/67. This is the proportion for Sample B.
  6. Find the difference between the two proportions: (42/67) minus (31/54).
  7. Determine a common denominator for 67 and 54. Since 67 is prime and 54 = 2×3^3, the least common denominator is 67×54 = 3618.
  8. Convert both fractions to have denominator 3618: (42×54)/3618 = 2268/3618, and (31×67)/3618 = 2077/3618.
  9. Subtract the numerators: 2268 - 2077 = 191.
  10. The difference is 191/3618. Check if this fraction can be simplified.
  11. Find the greatest common divisor (GCD) of 191 and 3618. 191 is a prime number (confirmed by testing divisibility up to sqrt(191) ≈ 13.8). 191 does not divide 3618 (3618 ÷ 191 ≈ 18.94, not integer), so GCD is 1.
  12. The simplified fraction is 191/3618. This is the exact difference between the proportions.

Reference answer: 191/3618

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