The birth weight and weight at age 21 of eight women are given in the table below. [table]
Birth weight (kg) | 1.9 | 2.4 | 2.6 | 2.7 | 2.9 | 3.2 | 3.4 | 3.6 Weight at 21 (kg) | 47.6 | 53.1 | 52.2 | 56.2 | 57.6 | 59.9 | 55.3 | 56.7
[/table] The value of the correlation coefficient is closest to:

Question

BlackTom AI · Accepted Answer

Pearson correlation coefficient r = Σ(xi - x̄)(yi - ȳ) / sqrt[Σ(xi - x̄)^2 · Σ(yi - ȳ)^2].

1) Relevant concepts
- The Pearson product-moment correlation measures linear association between two quantitative variables.
- Use sample means, deviations from means, cross-products and sums of squares to compute r.

2) Step-by-step reasoning and 3) intermediate calculations
- Data (n = 8):
  x (birth kg): 1.9, 2.4, 2.6, 2.7, 2.9, 3.2, 3.4, 3.6
  y (weight at 21 kg): 47.6, 53.1, 52.2, 56.2, 57.6, 59.9, 55.3, 56.7
- Compute means:
  x̄ = (sum x) / 8 = 22.7 / 8 = 2.8375
  ȳ = (sum y) / 8 = 438.6 / 8 = 54.825
- Compute deviations, cross-products and squares (showing summed results):
  Σ(xi - x̄)(yi - ȳ) = 11.6725
  Σ(xi - x̄)^2 = 2.17875
  Σ(yi - ȳ)^2 = 101.155
- Compute denominator:
  sqrt[2.17875 × 101.155] = sqrt(220.38770625) ≈ 14.8455
- Compute r:
  r = 11.6725 / 14.8455 ≈ 0.7863

4) Why the correct answer is correct
The calculated Pearson r ≈ 0.7863 matches option c. The algebraic steps above follow the exact definition of r using deviations and sums of squares; the arithmetic yields 0.7863, so option c is the closest.

5) Why other options are incorrect
- a (0.8232): too large; would imply a stronger linear association than the actual computed value.
- b (0.6182): substantially smaller; does not match the computed cross-product and sums of squares.
- d (0.536): much smaller and inconsistent with the positive, moderately strong association seen in the data.
Each of these alternatives deviates meaningfully from the precise calculation shown.

6) Conclusion / final confirmation
The Pearson correlation computed from the provided data is approximately 0.7863, so the correct choice is c. 0.7863.

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