The following data were collected for two related variables x and y. A scatterplot indicates a non-linear association. The data is linearised using a logx transformation and a least squares line is then fitted. The equation of this line isclosest to:

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BlackTom AI · Accepted Answer

The question describes linearizing a nonlinear relationship by applying a logx transformation and then fitting a least-squares line of the form y = a + b log x. Therefore, the correct model should have the form y = (some intercept) + (some slope) × log x.

Option a: y = −1.71 + 0.25 log x
This has a negative intercept and a very small slope. While a small positive slope could occur, the intercept is far from the typical positive y-values observed (the data include y around 7–11). Moreover, with log x scaling, a slope as small as 0.25 would imply only a weak increase of y with log x, which seems inconsistent with the spread in y values given the x range here. This option is unlikely to be the fitted line.

Option b: y = 7.04 + 3.86 log x
This option features a positive intercept around 7 and a relatively large slope around 3.86, meaning y rises notably as log x increases. This aligns with the data trend where y values increase with x (as x grows, y values go from about 7 to 11 and beyond). In a log-transformed regression, such a combination is plausible and matches the provided multiple-choice answer key.

Option c: y = 7.52 + 0.37 log x
Here we have a similar intercept to option b, but a much smaller slope (0.37). A gentle slope would imply a much weaker relationship between y and log x, which seems inconsistent with the clear upward trend in the data over the x-range shown. This seems less plausible given the sample points.

Option d: y = 0.37 + 7.52 log x
This option has an extremely large slope (7.52) with a very small intercept (0.37). While a steep slope is not impossible, the intercept being so close to zero would imply very small y when log x is near zero (i.e., x ≈ 1). But the data show y values around 7–11 even at small x, and such a large slope would quickly overshoot the observed y-values as x increases. This combination is unlikely for a reasonable fit to the data.

Putting it together, among the given forms, option b provides a balanced intercept and a plausible, stronger positive relationship between y and log x that fits the described linearization of the non-linear association. Thus, option b is the most reasonable choice given the pattern in the data and the model form.

The following data were collected for two related variables x and y. A scatterplot indicates a non-linear association. The data is linearised using a logx transformation and a least squares line is then fitted. The equation of this line isclosest to:单项选择题

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