A power regression model fits data using the form y = ax^b, where the response changes proportionally to a power of the explanatory variable; in AP Stats you linearize it by taking the logarithm of BOTH variables, then fit a least-squares regression line to the transformed data (Topic 2.9).
A power regression model describes a curved relationship with the equation y = ax^b. Think of geometry-style relationships, like area growing with the square of a side length or volume with the cube. When one variable changes proportionally to some power of the other, a straight line won't fit, and a power model will.
Here's the AP move. You don't actually fit a curve. You take the logarithm of both x and y, which turns y = ax^b into log(y) = log(a) + b·log(x). That's a linear equation, so now you can run an ordinary least-squares regression on the transformed data. If the residual plot for (log x, log y) looks random and r² jumps closer to 1, that's your evidence (per learning objective AP Stats 2.9.B) that the power model is the better fit. The slope of the transformed line is the exponent b, and the y-intercept is log(a).
Power regression lives in Topic 2.9, Analyzing Departures from Linearity, the last stop in Unit 2 (Exploring Two-Variable Data). It directly supports learning objective AP Stats 2.9.B, which asks you to calculate a predicted response using a least-squares regression line for transformed data. The CED's essential knowledge is explicit about the evidence you cite: increased randomness in the residual plot after transformation and r² moving closer to 1 both signal that the transformed model is more appropriate than the untransformed one. The big conceptual payoff is that everything you learned about LSRLs, slope, intercept, residuals, and r² still applies. You're just applying it to logged variables, then undoing the transformation when you make a prediction.
Keep studying AP Statistics Unit 2
Exponential Regression (Unit 2)
These are the two transformation models in Topic 2.9, and the difference comes down to which variables get logged. Exponential models (y = ab^x) only need log(y) versus x. Power models (y = ax^b) need log(y) versus log(x). If logging both variables straightens the scatterplot but logging only y doesn't, you've got a power relationship.
Residuals (Unit 2)
The residual plot is your verdict on whether a power model was the right call. A curved residual pattern on the original data says the linear model fails; a random scatter of residuals after the log-log transformation says the power model works. You read the same plot, just on transformed data.
Influential Point (Unit 2)
Topic 2.9 pairs transformations with influential points under the same learning objective family (AP Stats 2.9.A). A high-leverage point can drag the slope of your transformed regression just like it drags a regular LSRL, so check for unusual points before and after transforming.
y-intercept (Unit 2)
In a log-log regression, the y-intercept isn't a itself, it's log(a). To recover the actual coefficient of the power model, you have to raise 10 (or e, depending on the log used) to the intercept. Forgetting this back-transformation is one of the most common prediction errors.
No released FRQ has used the phrase 'power regression model' verbatim, but transformed regression is fair game under Topic 2.9, and it usually shows up like this: you're handed computer output for a regression of log(y) on log(x) and asked to write the equation, make a prediction, or justify why the transformed model beats the original. Multiple-choice stems often show two residual plots or two r² values and ask which model is more appropriate. The traps to avoid are predictable. Plug the x-value into the transformed equation correctly (you need log(x), not x), and remember the equation outputs log(y), so you must exponentiate to get the actual predicted y. Justifications should cite the CED's two pieces of evidence: a more random residual plot and an r² closer to 1.
Both fix curved data with logarithms, but they log different things. An exponential model (y = ab^x) becomes linear when you take log(y) only, so you plot log(y) against x. A power model (y = ax^b) becomes linear only when you take the log of BOTH variables, so you plot log(y) against log(x). Quick check: in an exponential model the exponent is the variable x; in a power model the exponent is the constant b. On the exam, look at the axis labels in the computer output. If both axes are logged, it's a power model.
A power regression model has the form y = ax^b, where the response variable changes proportionally to a power of the explanatory variable.
To fit a power model in AP Stats, take the logarithm of both x and y, then run a regular least-squares regression on the transformed data.
The transformed equation log(y) = log(a) + b·log(x) means the LSRL slope is the exponent b and the intercept is log(a), not a itself.
Evidence that the power model fits better is a more random residual plot after transformation and an r² value closer to 1, exactly as stated in the CED for Topic 2.9.
To make a prediction, plug log(x) into the transformed equation to get log(y), then exponentiate to find the actual predicted y.
Power models log both variables, while exponential models log only y; checking which transformation straightens the scatterplot tells you which model to use.
It's a model of the form y = ax^b used when the response variable changes proportionally to a power of the explanatory variable. In AP Stats (Topic 2.9), you handle it by logging both variables and fitting a least-squares regression line to (log x, log y).
No. The AP exam never asks you to fit a curve directly. You transform the data with logarithms so it becomes linear, then use the same LSRL tools from earlier in Unit 2. Learning objective AP Stats 2.9.B only requires predictions from a least-squares line on transformed data.
An exponential model (y = ab^x) is linearized by logging only the y-variable, while a power model (y = ax^b) requires logging both x and y. If the computer output shows a regression of log(y) on log(x), you're looking at a power model.
The CED gives two pieces of evidence: the residual plot for the transformed data shows more random scatter than the original, and r² moves closer to 1. Cite both when justifying your model choice on an FRQ.
Take the log of your x-value, plug it into the transformed equation to get a predicted log(y), then exponentiate (raise 10 or e to that value) to get the actual prediction. Forgetting the final back-transformation step gives you log(y) instead of y, a classic point-loser.
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