# How would I test to see if a data set is described by a specified (not calculated) 2nd order polynomial line?

In Short: Given a set of data and a specified 2nd order polynomial trend line, how can I go about testing a hypothesis that the line describes the data?

I'm not asking to do a linear regression on the data to find the line of best fit with associated statistics. I want to determine if a predetermined trend line is likely to describe the data.

In Long: I'm attempting to verify a simulation model. The goal is to produce simulated data that follows the same characteristics of an original data set. This set is composed of introduction dates and lifetimes.

Information about the original set is limited. The lifetimes follow a Weibull distribution. Shape and scale parameters for the collective data set of lifetimes are given. However, the mean lifetime changes over time (dependent on the introduction date). This change in mean lifetime is described by a specified trend line. All we know about the trend line are its coefficients (no r-squared, F, or p -values). The distribution of introduction dates is unknown.

Currently the simulation runs like this:

• Assuming uniformly distributed introduction dates, generate dates
• Calculate a mean lifetime for introduction dates based on the given trend line
• Assume and calculate a constant standard deviation (calculated from the original collective shape and scale parameters)
• Using the constant standard deviation and the calculated mean lifetimes, solve a non-linear system of equations to find shape and scale parameters for each introduction date
• Use these shape and scale parameters to specify a Weibull distribution from which to draw a simulated lifetime.

Generating 100 simulated points in this manner and plotting them with the original trend line produces something like the following:

Given the limitations, I feel the best I can do in regards to verification is to see if the original trend line describes the data with some sort of statistical test and compute the parameters for the collective simulated lifetimes to see if the confidence intervals contain the original parameters.

I'm working mostly in R, so references to R packages, or strategies in R would be appreciated.