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:Single choice
A
a. y = 7.04 + 3.86 log x
B
b. y = −1.71 + 0.25 log x
C
c. y = 0.37 + 7.52 log x
D
d. y = 7.52 + 0.37 log x

Question

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:Single choice

A

a. y = 7.04 + 3.86 log x

B

b. y = −1.71 + 0.25 log x

C

c. y = 0.37 + 7.52 log x

D

d. y = 7.52 + 0.37 log x

BlackTom AI · Accepted Answer

First, consider the transformation described: a logx transformation is applied to x, and then a least-squares line is fitted to y versus log x. This means the model is of the form y = a + b log(x), where log is typically base 10 in many introductory analyses unless stated otherwise.

Option a: y = 7.04 + 3.86 log x
- This has an intercept around 7.0 and a positive slope close to 3.9, which matches the observed pattern: when log x is near 0 (x close to 1), y is near 7; as log x increases toward about 1 (x near 10), y increases toward ~11. This aligns well with the data trend.

Option b: y = −1.71 + 0.25 log x
- The intercept is far too small (negative), and the slope is much too shallow. With log x ranging roughly from 0 to 1, this would imply y changes by about 0.25, hardly capturing the observed rise from about 7 to ~11. This is inconsistent with the data pattern.

Option c: y = 0.37 + 7.52 log x
- Here the intercept is near 0.37, which would place y near 0.37 when log x is 0 (x = 1). However the actual y at x = 1 is 7, so this intercept is clearly implausible given the data. Moreover, a slope as large as 7.52 would imply very steep growth as log x increases, which would overestimate y for larger x compared to the observed values.

Option d: y = 7.52 + 0.37 log x
- This has a plausible intercept around 7.5, which could align with y near 7 when x ≈ 1. However, the slope is only 0.37, which would yield a very small increase in y across the log x range (0 to about 1), predicting y around 7.4 to 7.9, far below the observed values near 11 as x grows. This underestimates the growth shown in the data.

In summary, option a provides a reasonable intercept near 7 and a slope near 3.9, which together reproduce the gradual, substantial increase in y as log x increases from 0 to about 1, matching the observed y-values across x = 1 to 11. The other options either place the line too low or too high, or assign an implausible rate of increase that disagrees with the data pattern.

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:Single choice

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