You wish to fit a linear relationship to the points listed below, where the points have the format (xi, yi): (1.2, 7.9)(1.8, 6.6)(2.4, 5.9)(2.8, 5.5)(3.2, 4.6)(3.8, 4.1) Using a Python function of your choice, determine the gradient, y-intecept and coefficient of determination, all to 4 decimal places. These are...

Question

BlackTom AI · Accepted Answer

Linear least-squares regression uses slope b = S_xy / S_xx and intercept a = ȳ − b x̄, with S_xy = Σ(xi − x̄)(yi − ȳ), S_xx = Σ(xi − x̄)^2 and R^2 = (S_xy)^2/(S_xx·S_yy).

1) Relevant knowledge points and formulas
- Mean: x̄ = Σxi / n, ȳ = Σyi / n.
- Slope: b = S_xy / S_xx.
- Intercept: a = ȳ − b x̄.
- Coefficient of determination: R^2 = (S_xy)^2/(S_xx·S_yy) where S_yy = Σ(yi − ȳ)^2.

2) Step-by-step computation
Given points: (1.2,7.9),(1.8,6.6),(2.4,5.9),(2.8,5.5),(3.2,4.6),(3.8,4.1). n = 6.

Compute means:
Σx = 15.2 → x̄ = 15.2/6 = 2.5333333333
Σy = 34.6 → ȳ = 34.6/6 = 5.7666666667

Compute deviations and sums:
For each point compute dx = xi − x̄ and dy = yi − ȳ, then dx·dy and dx^2, dy^2.

Calculated sums:
S_xy = Σ(dx·dy) = −6.4333333333
S_xx = Σ(dx^2) = 4.4524444444
S_yy = Σ(dy^2) = 9.4733333333

(Intermediate values shown above come from summing the six individual dx·dy, dx^2 and dy^2 terms.)

3) Calculations (with intermediate steps)
Slope: b = S_xy / S_xx = −6.4333333333 / 4.4524444444 = −1.444558... → rounded to 4 d.p.: −1.4446

Intercept: a = ȳ − b x̄ = 5.7666666667 − (−1.444558...)(2.5333333333)
= 5.7666666667 + 3.659633... = 9.426300... → rounded to 4 d.p.: 9.4263

R^2: = (S_xy)^2 / (S_xx · S_yy)
= (−6.4333333333)^2 / (4.4524444444 · 9.4733333333)
= 41.403... / 42.179... = 0.9810 (rounded to 4 d.p.)

4) Why this answer is correct
The numbers come directly from the standard least-squares formulas and the explicit sums computed from the six given points. Rounding each final value to four decimal places yields slope = −1.4446, intercept = 9.4263, and R^2 = 0.9810.

5) Why the other options are incorrect (brief)
- Options a, b, c, g, h give positive slopes that contradict the clear downward trend of y vs x.
- Options d and f give wildly different intercepts or slopes inconsistent with the computed sums and produce poorer fits (their R^2 values and magnitudes do not match the actual sums).
- Only option e matches the signs, magnitudes and the computed R^2 from the least-squares calculation.

6) Conclusion
The computed gradient, y-intercept and coefficient of determination (rounded to 4 decimal places) are gradient = −1.4446, y-intercept = 9.4263, r2 = 0.9810, which confirms option e.

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