NavGanitMateby NavGreen Compute, AustraliaBased on Australian Curriculum: Mathematics v9.0
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Sampling methods and bias

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Required skills (click to reveal)

A library in Brisbane uses a systematic sampling method to survey patron preferences for e-book access. The library has 940 regular patrons and selects every 14th patron from a list ordered by membership number.

If the first patron selected is number 9, what is the membership number of the 10th patron selected? Express your answer as an exact integer.

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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.
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.

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Explanation

Worked solution and answer (click to reveal)

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Worked solution

  1. The library uses systematic sampling, selecting every 14th patron from a list ordered by membership number.
  2. The first patron selected has membership number 9.
  3. To find the 10th patron selected, we need to determine the 10th term in an arithmetic sequence starting at 9 with a common difference of 14.
  4. The general formula for the nth term of an arithmetic sequence is: a_n = a_1 + (n - 1) * d.
  5. Substitute a_1 = 9, d = 14, and n = 10 into the formula: a_10 = 9 + (10 - 1) * 14.
  6. Simplify the expression inside the parentheses: (10 - 1) = 9.
  7. Multiply: 9 * 14 = 126.
  8. Add: 9 + 126 = 135.
  9. The 10th patron selected has membership number 135.
  10. Verify that 135 is within the library’s patron list (940 patrons), so it is a valid membership number.
  11. Check the pattern: 9, 23, 37, 51, 65, 79, 93, 107, 121, 135. Each step adds 14, confirming the 10th term is correct.
  12. The membership number of the 10th patron selected is 135.

Reference answer: 135

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