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 association between x and y that becomes linear after applying a log transformation to x, and then fitting a least-squares line to predict y from log x. To evaluate the candidate equation, consider what a linear relation y = a + b log10(x) implies:
- When x = 1, log10(x) = 0, so y should be approximately the intercept a. The observed y at x = 1 is 7, which suggests the intercept is around 7 if the model is reasonably accurate.
- As x increases, log10(x) increases slowly, so the slope b controls how quickly y rises with x on the log-scale. A positive b means y grows with x, which matches the general upward trend in the data.
- For larger x, like x = 11, log10(11) ≈ 1.041. If a ≈ 7 and b ≈ 3.86, the predicted y is 7 + 3.86 × 1.041 ≈ 7 + 4.02 ≈ 11.0, which aligns well with the observed y values around 11.1. This consistency across the range supports the plausibility of the given form.
Now evaluate the provided option:
- Option c: y = 7.04 + 3.86 log x. This matches the idea that the intercept is near 7 and the slope is a positive number that, when multiplied by log10(x), yields predictions that track the ascending y-values as x increases. The numbers 7.04 and 3.86 are reasonable magnitudes given the data points (y ranges roughly from 7 to about 11).
- If there were other options (not shown here), we would test them similarly: compare predictive behavior at small x (where log x ≈ 0) to the intercept, examine whether the slope sign and magnitude produce the observed gradual rise, and check predictions at the largest x against the highest y values.
In summary, the proposed equation y = 7.04 + 3.86 log x captures the key features of a linear relationship when x is transformed by a logarithm: an intercept around 7 when x is small, a positive slope producing a steady increase in y with increasing log x, and predicted values that align with the observed y-values across the x-range. This alignment with the data pattern explains why this form is a plausible least-squares fit for log x versus y.

MUF0092 Mathematics Unit 2-SEM 1-2025 14.7 Study Area 3 Revision

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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