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)

In a walking-route study in Melbourne, two random samples of 95 and 39 participants were taken to compare the proportion who completed a 12 km route within 2 hours. In the first sample of 95 participants, 57 completed the route.

In the second sample of 39 participants, 23 completed the route. What is the exact difference between the proportion of participants who completed the route in the first sample and the second sample?

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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Formula

Explanation

Worked solution and answer (click to reveal)

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Worked solution

  1. The first sample has 95 participants, of which 57 completed the 12 km route within 2 hours. The proportion is calculated as 57 divided by 95.
  2. Compute the proportion for the first sample: 57 / 95 = 0.6. This is exact because 57 ÷ 95 = 3/5 after simplifying by dividing numerator and denominator by 19.
  3. The second sample has 39 participants, of which 23 completed the route. The proportion is 23 / 39.
  4. Compute the proportion for the second sample: 23 / 39. This fraction cannot be simplified further as 23 is prime and does not divide 39.
  5. Find the difference between the two proportions: (57 / 95) - (23 / 39).
  6. Simplify 57 / 95: divide both by 19 to get 3 / 5.
  7. The difference is now (3 / 5) - (23 / 39).
  8. Find a common denominator for 5 and 39. The least common multiple of 5 and 39 is 195.
  9. Convert both fractions to have denominator 195: (3 / 5) = (3 × 39) / 195 = 117 / 195; (23 / 39) = (23 × 5) / 195 = 115 / 195.
  10. Subtract: 117 / 195 - 115 / 195 = 2 / 195.
  11. The fraction 2 / 195 is already in simplest form because 2 and 195 share no common factors other than 1.
  12. The exact difference between the proportions is 2 / 195.

Reference answer: 2/195

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