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 Perth claims that 76 out of every 100 people in a sample of 500 residents have used a new government mental health support service. The report states this rate is representative of the entire adult population in Western Australia.

What is the exact proportion of respondents who used the service, expressed as a simplified fraction?

Concept illustrations

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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 76 out of every 100 people in a sample of 500 residents used the mental health service. This means 76% of the 500 respondents used the service.
  2. To find the exact number of people who used the service, calculate 76% of 500. Convert 76% to a decimal by dividing by 100: 76 ÷ 100 = 0.76.
  3. Multiply the decimal by the sample size: 0.76 × 500 = 380. So, 380 people used the service.
  4. The proportion of respondents who used the service is 380 out of 500. This is written as the fraction 380/500.
  5. Simplify the fraction 380/500 by finding the greatest common divisor (GCD) of 380 and 500. First, factor both numbers: 380 = 2×2×5×19, 500 = 2×2×5×5×5.
  6. The common prime factors are 2×2×5 = 20. Divide both numerator and denominator by 20: 380 ÷ 20 = 19, 500 ÷ 20 = 25.
  7. The simplified fraction is 19/25. This is the exact proportion of respondents who used the service.
  8. Check the simplification: 19 and 25 have no common factors other than 1, so 19/25 is in simplest form.
  9. Verify the original value: 19/25 = 0.76, which is 76%, matching the given rate.
  10. The proportion is correctly expressed as a simplified fraction: 19/25.

Reference answer: 19/25

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