Fit linear, exponential, and other regression models to data; interpret the parameters and the correlation coefficient in context; use a residual plot to judge whether a model is appropriate; and use a model to make predictions.

What this topic is asking

The Regents Algebra II exam (the Interpreting Data, S-ID, and Making Inferences, S-IC, clusters) wants you to fit regression models (linear, exponential, and others), interpret their parameters and the correlation coefficient in context, use a residual plot to judge whether a model is appropriate, and use a model to make predictions. This extends the Algebra I line of best fit to nonlinear models and adds the residual-plot diagnostic.

Fitting a regression model

Regression finds the function that best fits a data set. Algebra II extends beyond the linear case to exponential and other models, all produced by a graphing calculator's regression functions. You interpret the parameters in context just as with a line of best fit:

• Linear $\hat{y} = mx + b$: slope $m$ is the predicted change in $y$ per unit of $x$; intercept $b$ is the predicted value at $x = 0$.
• Exponential $\hat{y} = ab^x$: $a$ is the initial value and $b$ is the growth factor (decay if $0 < b < 1$).

The correlation coefficient

The correlation coefficient $r$ measures how well a linear model fits.

So $r = -0.95$ is a strong negative linear relationship, and $r = 0.1$ is essentially no linear relationship. As in Algebra I, a strong $r$ supports prediction but never proves causation.

Residual plots

A residual is actual minus predicted, and a residual plot graphs these residuals against the $x$-values. It is the key diagnostic for whether a model is the right type.

Predicting and reasoning for credit

Once a model is chosen, predict by substituting an $x$-value into the equation. A clarifying point worth stressing is that the residual plot, not just $r$, decides the model type: a fairly high $r$ can still accompany a curved residual pattern that reveals a linear model is wrong for nonlinear data. The Regents commonly pairs a prediction (substitute and evaluate) with an explanation of how a residual plot supports the chosen model, and the explanation credits require describing the pattern (random scatter good, systematic pattern bad). A second habit is to keep interpreting the correlation coefficient correctly: magnitude is strength, sign is direction, and correlation is never causation.

Try this

Q1. A regression has $r = 0.88$. Describe the relationship. [1 credit]

• Cue. Strong positive linear relationship (magnitude near 1, positive sign).

Q2. An exponential model is $\hat{y} = 20(1.5)^x$. Predict $y$ at $x = 3$. [2 credits]

• Cue. $20(1.5)^3 = 20(3.375) = 67.5$.