NavGanitMateby NavGreen Compute, AustraliaBased on Australian Curriculum: Mathematics v9.0
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Evaluating statistical claims

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A statistical report in Sydney claims that 44 out of every 41 people surveyed in a metropolitan area have used a new public transport app. The report states this proportion is representative of the entire city population.

What is the exact value of the proportion of surveyed people who used the app, expressed as a simplified fraction?

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Median, mode and range in sorted data. The ordered values are 2,3,3,8,9. The middle value is 3, the most frequent value is 3, and the range is 9 − 2 = 7. The row uses equal spacing, not a number-line scale.23389↑ medianmode 3 range 7
The ordered values are 2,3,3,8,9. The middle value is 3, the most frequent value is 3, and the range is 9 − 2 = 7. The row uses equal spacing, not a number-line scale.
Comparing three category counts. Illustrative category counts: A = 2, B = 4, C = 9. Separate bars compare categories; the vertical scale starts at zero.05102A4B9C5
Illustrative category counts: A = 2, B = 4, C = 9. Separate bars compare categories; the vertical scale starts at zero.

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Worked solution

  1. The report states that 44 out of every 41 people surveyed have used the app. This is given as a ratio of people who used the app to the total surveyed.
  2. The proportion is expressed as a fraction: 44 / 41.
  3. To simplify the fraction, determine the greatest common divisor (GCD) of 44 and 41.
  4. 41 is a prime number and does not divide 44 evenly. The only common factor is 1.
  5. Since the GCD of 44 and 41 is 1, the fraction 44/41 is already in its simplest form.
  6. The proportion of surveyed people who used the app is therefore 44/41.
  7. This fraction is greater than 1, which means more people used the app than were surveyed, which is mathematically impossible.
  8. The claim that 44 out of 41 people used the app is logically inconsistent and cannot represent a real proportion.
  9. Therefore, the proportion cannot be expressed as a valid simplified fraction because the data is impossible.
  10. The scenario presents a statistical error: a proportion cannot exceed 1 (or 100%) in reality.
  11. The fraction 44/41 is mathematically valid but represents an impossible situation, so it cannot be a correct proportion of people.
  12. The exact value of the proportion is 44/41, but it is not a valid proportion for a real survey.

Reference answer: 44/41

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