Practice problems for these concepts can be found at:

- Two-Variable Data Analysis Multiple Choice Practice Problems for AP Statistics
- Two-Variable Data Analysis Free Response Practice Problems for AP Statistics
- Two-Variable Data Analysis Review Problems for AP Statistics
- Two-Variable Data Analysis Rapid Review for AP Statistics

Until now, we have been concerned with data that can be modeled with a line. Of course, there are many two-variable relationships that are nonlinear. The path of an object thrown in the air is parabolic (quadratic). Population tends to grow exponentially, at least for a while. Even though you could find a LSRL for nonlinear data, it makes no sense to do so. The AP Statistics course deals only with two-variable data that can be modeled by a line OR non-linear two-variable data that can be *transformed* in such a way that the transformed data can be modeled by a line.

example:Letg(x) = 2x, which is exponential and clearly nonlinear. Letf(x) =ln(x). Then,f[g(x)] =ln(2x) =xln(2), which is linear. That is, we can transform an exponential function such asg(x) into a linear function by taking the log of each value ofg(x).

example:Letg(x) = 4x^{2}, which is quadratic. Letf(x) = . Thenf[g(x)] = =2x, which is linear.

example:The number of a certain type of bacteria present (in thousands) after a certain number of hours is given in the following chart:

What would be the predicted quantity of bacteria after 3.75 hours?

solution:A scatterplot of the data and a residual plot [forNumber=a+b(Hour)] shows that a line is not a good model for this data:

Now, take *ln*(*Number*) to produce the following data:

The scatterplot of *Year* versus *ln*(*Population*) and the residual plot for *ln*(*Number*) = –0.0047 + 0.586(*Hours*) are as follows:

The scatterplot looks much more linear and the residual plot no longer has the distinctive pattern of the raw data. We have transformed the original data in such a way that the transformed data is well modeled by a line. The regression equation for the transformed data is: *ln*(*Number*) = –0.047 + 0.586 (*Hours*).

The question asked for how many bacteria are predicted to be present after 3.75 hours. Plugging 3.75 into the regression equation, we have *ln*(*Number*) = –0.0048 + 0.586(3.75) = 2.19. But that is *ln*(*Number*), not *Number*. We must back-transform this answer to the original units. Doing so, we have *Number* = *e*^{2.19} = 8.94 thousand bacteria.

It may be worth your while to try several different transformations to see if you can achieve linearity. Some possible transformations are: take the log of both variables, raise one or both variables to a power, take the square root of one of the variables, take the reciprocal of one or both variables, etc.

Practice problems for these concepts can be found at:

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