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
A
a. y = −1.71 + 0.25 log x
B
b. y = 7.04 + 3.86 log x
C
c. y = 7.52 + 0.37 log x
D
d. y = 0.37 + 7.52 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:单项选择题

A

a. y = −1.71 + 0.25 log x

B

b. y = 7.04 + 3.86 log x

C

c. y = 7.52 + 0.37 log x

D

d. y = 0.37 + 7.52 log x

BlackTom AI · Accepted Answer

To approach this question, we first note the situation: two related variables x and y show a non-linear relationship, but after applying a log transformation to x (log x) the relationship becomes linear, and a least-squares line is fitted to y versus log x. The task is to identify which linear equation in log x best fits the data.

Option a: y = −1.71 + 0.25 log x
- The slope here is very small (0.25), implying y would increase very slowly as log x increases. Given that y values in the table range roughly from 7 to 11 and log x spans a modest range for x from 1 to 11, such a tiny slope would not capture the observed rise in y as x grows. This makes option a unlikely to be the best fit.

Option b: y = 7.04 + 3.86 log x
- This option has a much larger slope (3.86), meaning y would rise quite noticeably with log x. The intercept 7.04 places baseline y near the lower end of the observed y values, while the steep slope allows y to reach around 11 as log x increases, which aligns with the data progression from x ≈ 1 to x ≈ 11. Conceptually, this is a plausible linear relationship after log-transforming x, producing the rapid y increases observed.

Option c: y = 7.52 + 0.37 log x
- With a slope of 0.37, this is a very shallow increase of y with log x. The actual data show y rising by about 4 units over the x range, which would require a substantially larger slope to approximate the trend. This option underestimates the effect of log x on y.

Option d: y = 0.37 + 7.52 log x
- Here the slope is extremely large (7.52). This would cause y to rise very quickly with log x, potentially producing values far beyond the observed y range for moderate log x, and would typically overshoot the data unless log x is tiny. Given the observed y values, this slope is unrealistically steep.

In summary, among the four linear models in log x, options a, c, and d show either too shallow, too steep, or implausible intercept-slope combinations to match the observed data pattern. The model with a moderate-to-strong slope that can reasonably map y across the range of log x values—y = 7.04 + 3.86 log x—best aligns with the data behavior after log-transforming x. Based on this evaluation, option b is the most consistent with the described linearization and 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:单项选择题

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