Asked by 123
Charlie used a regression calculator to generate the equation f(x) = –0.15x + 20.1 for the ordered pairs (2, 15), (4, 21), (6, 26), (8, 20), and (10, 14).
Is a linear representation the best way to represent the data? If it is, explain why. If not, explain why and suggest a better alternative.
All Answers 1
Answered by
GPT-5 mini
AI
No — a linear model is not the best representation.
Why: the y-values rise from x = 2 to x = 6 (15 → 26) and then fall from x = 6 to x = 10 (26 → 14). The residuals from the reported linear fit (f(2)=19.8 vs 15, f(6)=19.2 vs 26, f(10)=18.6 vs 14) show a systematic pattern (negative, large positive, negative) rather than random scatter, which indicates curvature in the data that a straight line cannot capture.
Better alternative: a quadratic (parabolic) model fits the rise-and-fall pattern. For example, a parabola with vertex at (6, 26),
y = −0.6875(x − 6)^2 + 26
(or expanded y = −0.6875x^2 + 8.25x + 1.25),
matches (2,15) exactly and gives much smaller residuals overall than the linear fit. A quadratic (or generally a nonlinear regression) is therefore a better choice.
Why: the y-values rise from x = 2 to x = 6 (15 → 26) and then fall from x = 6 to x = 10 (26 → 14). The residuals from the reported linear fit (f(2)=19.8 vs 15, f(6)=19.2 vs 26, f(10)=18.6 vs 14) show a systematic pattern (negative, large positive, negative) rather than random scatter, which indicates curvature in the data that a straight line cannot capture.
Better alternative: a quadratic (parabolic) model fits the rise-and-fall pattern. For example, a parabola with vertex at (6, 26),
y = −0.6875(x − 6)^2 + 26
(or expanded y = −0.6875x^2 + 8.25x + 1.25),
matches (2,15) exactly and gives much smaller residuals overall than the linear fit. A quadratic (or generally a nonlinear regression) is therefore a better choice.
Submit Your Answer
We prioritize human answers over AI answers.
If you are human, and you can answer this question, please submit your answer.