A set of bivariate data involving the variables [math: x] and [math: y] undergoes a statistical analysis. A summary of that analysis is shown in the table below.The equation of the least squares regression line is [math: y=5.14−1.25x].The value of the Pearson's correlation coefficient is closest to

Question

BlackTom AI · Accepted Answer

Slope of the least squares line is related to Pearson’s r by b = r * (sy / sx), so r = b * (sx / sy).

1) Relevant knowledge points and concepts
- The slope b of the regression line of y on x equals r · (sy/sx), where r is Pearson’s correlation, sx and sy are the sample standard deviations of x and y respectively.
- The sign of r matches the sign of the slope: a negative slope implies a negative correlation.
- Numerical rounding may be required to match the multiple-choice options.

2) Step-by-step reasoning
- We are given the regression line y = 5.14 − 1.25x, so b = −1.25.
- From the table: sx = 6.4 and sy = 8.4.
- Use r = b · (sx / sy).

3) Calculations with intermediate steps
- Compute the ratio sx/sy = 6.4 / 8.4.
  Simplify: 6.4/8.4 = 64/84 = 16/21 ≈ 0.76190476.
- Multiply by the slope: r = −1.25 × 0.76190476.
  Note that −1.25 = −5/4, so r = −(5/4) × (16/21) = −80/84 = −20/21.
- Evaluate −20/21 ≈ −0.95238095, which rounds to −0.95.

4) Why this answer is correct
- The formula r = b(sx/sy) is the exact relationship between slope and correlation for standardized units. Plugging the given slope and standard deviations yields r = −20/21 ≈ −0.952, which is closest to −0.95.

5) Why the other options are incorrect
- −0.70 and 0.70 are too small in magnitude compared with the computed −0.952.
- 0.95 has the wrong sign (positive) while the slope is negative; the correct r must be negative.

6) Conclusion
- The Pearson correlation coefficient is closest to −0.95, matching option A.

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