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

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Required skills (click to reveal)

A statistical report in Adelaide claims that 36% of residents in a regional council area have experienced food insecurity in the past year. The report is based on a survey of 840 randomly selected households.

The survey found that 302 households reported food insecurity. Evaluate the claim by calculating the exact proportion of households in the sample that reported food insecurity, 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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Explanation

Worked solution and answer (click to reveal)

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Worked solution

  1. The survey sampled 840 households, and 302 reported food insecurity. The proportion is calculated as the number of households reporting food insecurity divided by the total number of households surveyed.
  2. Set up the fraction: 302 / 840.
  3. Simplify the fraction by finding the greatest common divisor (GCD) of 302 and 840. First, factor 302: 302 = 2 × 151. Factor 840: 840 = 2 × 2 × 2 × 3 × 5 × 7.
  4. The only common factor between 302 and 840 is 2. Divide both numerator and denominator by 2.
  5. 302 ÷ 2 = 151. 840 ÷ 2 = 420.
  6. The simplified fraction is 151 / 420.
  7. Check if 151 and 420 have any common factors. 151 is a prime number (it is not divisible by 2, 3, 5, 7, 11, or 13). 420 is not divisible by 151.
  8. Since 151 is prime and does not divide 420, the fraction 151/420 is in its simplest form.
  9. The exact proportion of households reporting food insecurity is 151/420.
  10. Verify by decimal conversion: 151 ÷ 420 ≈ 0.3595, which rounds to 36% when expressed as a percentage. This matches the claim.
  11. The claim states 36% of residents experienced food insecurity. The sample proportion is 151/420, which is the exact simplified fraction.
  12. The exact proportion of households in the sample that reported food insecurity, expressed as a simplified fraction, is 151/420.

Reference answer: 151/420

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