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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The problem describes a non-linear relationship between x and y that becomes linear after applying a log base transformation to x, and then a least-squares line is fitted to the transformed data. In this framework, the model has the form y = α + β log x, where α is the intercept and β is the slope with respect to log x.

Option a. y = −1.71 + 0.25 log x
This gives a very small slope (0.25) and a negative intercept. Given the data trend and the transformation described, such a tiny slope would imply only a very weak increase of y with log x, which is inconsistent with the typical scale of y values shown in the scatter (and with the pre-known correct form in option c). Therefore this option is unlikely to be the least-squares line after log x transformation.

Option b. y = 0.37 + 7.52 log x
Here the slope is extremely large (7.52), which would produce a very steep increase of y with log x, likely overshooting the observed y values across the x range. This mismatch in scale suggests this is not the best-fitting line for these data.

Option c. y = 7.04 + 3.86 log x
This option has a moderate intercept around 7 and a positive slope around 3.86, which is reasonable given the y-values in the table (roughly 7 to 11) and the transformation on x. It aligns with the stated outcome as the closest among the provided choices to the least-squares fit for log x, matching the typical pattern of a linear relationship after log transformation.

Option d. y = 7.52 + 0.37 log x
This combination features a larger intercept (7.52) but a very small slope (0.37). Such a flat response to log x would not capture the apparent variation in y across the range of x in the dataset, making it unlikely to be the fitted line for a clear linear trend after log transformation.

In summary, the model form implied by the problem is y = α + β log x, and among the given choices, option c provides a plausible intercept and slope that reflect a balanced fit to the data after log x transformation. The other options either misrepresent the scale of the slope or the intercept, leading to inconsistent predictions across the x range.

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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