NavGanitMateby NavGreen Compute, AustraliaBased on Australian Curriculum: Mathematics v9.0
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Bivariate statistical investigations

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Required skills (click to reveal)

A school in Sydney conducts a statistical investigation into the relationship between the number of textbooks ordered (x) and the total cost (y in AUD). The collected bivariate data is: (54, 97), (56, 101), (58, 105), (60, 109), (62, 113).

Using linear regression, determine the exact value of the y-intercept of the line of best fit. Express your answer as a simplified fraction.

Concept illustrations

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Association between paired values. Authored example pairs show a positive association: larger x tends to accompany larger y. A scatterplot alone cannot establish causation. Axes show arbitrary numerical units.xyPaired observations
Authored example pairs show a positive association: larger x tends to accompany larger y. A scatterplot alone cannot establish causation. Axes show arbitrary numerical units.
A straight-line relationship. Illustration: y = x + 1. Each unit right adds one unit up; the y-intercept is 1.-2-2-1-111223344xy
Illustration: y = x + 1. Each unit right adds one unit up; the y-intercept is 1.

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Worked solution and answer (click to reveal)

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Worked solution

  1. The bivariate data consists of textbook orders (x) and total cost in AUD (y). The data points are: (54, 97), (56, 101), (58, 105), (60, 109), (62, 113). We are to find the y-intercept of the line of best fit using linear regression.
  2. Linear regression uses the formula for the line of best fit: y = mx + b, where m is the slope and b is the y-intercept. We need to compute b using the formula: b = (Σy)(Σx²) - (Σx)(Σxy) / [n(Σx²) - (Σx)²].
  3. Calculate the sum of x values: Σx = 54 + 56 + 58 + 60 + 62 = 290.
  4. Calculate the sum of y values: Σy = 97 + 101 + 105 + 109 + 113 = 525.
  5. Calculate the sum of x²: 54² = 2916, 56² = 3136, 58² = 3364, 60² = 3600, 62² = 3844. Σx² = 2916 + 3136 + 3364 + 3600 + 3844 = 16860.
  6. Calculate the sum of xy: 54×97 = 5238, 56×101 = 5656, 58×105 = 6090, 60×109 = 6540, 62×113 = 7006. Σxy = 5238 + 5656 + 6090 + 6540 + 7006 = 30530.
  7. n = 5 (number of data points).
  8. Substitute into the formula for b: b = [Σy × Σx² - Σx × Σxy] / [n × Σx² - (Σx)²].
  9. Numerator: (525 × 16860) - (290 × 30530). First, 525 × 16860 = 8,851,500. Second, 290 × 30530 = 8,853,700. So numerator = 8,851,500 - 8,853,700 = -2,200.
  10. Denominator: 5 × 16860 - (290)² = 84,300 - 84,100 = 200.
  11. b = -2200 / 200 = -11. The y-intercept is -11.
  12. The y-intercept of the line of best fit is -11. This means that if no textbooks were ordered, the cost would be -11 AUD, which is not realistic but mathematically valid in the model.

Reference answer: -11

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