Two-Variable Data: Models and Scatterplots

Two-variable data pairs each $x$-value with a $y$-value, usually shown in a scatterplot or a table. The SAT asks you to describe the association, use a line of best fit to make predictions, interpret the slope and intercept in context, compute residuals, and decide whether a linear or exponential model fits the data better.

• Direction: points rising left to right show a positive association; falling shows a negative association.
• Strength: points close to a line show a strong association; a loose cloud shows a weak one.
• A line of best fit gives predicted values, not exact ones. Plug the $x$-value into the equation to predict $y$.
• The slope is the predicted change in $y$ for each increase of $1$ in $x$. The $y$-intercept is the predicted $y$ when $x = 0$, which may not make sense in context.
• A residual is $\text{actual} - \text{predicted}$. A positive residual means the point lies above the line.
• Linear data changes by a constant difference; exponential data changes by a constant ratio.

• "Starts at $a$ and triples every $k$ years" is $y = a \cdot 3^{x/k}$.
• Line through two points: slope $m = \frac{y_2 - y_1}{x_2 - x_1}$, then solve for $b$.

Example: $x = 0, 1, 2, 3$ with $y = 5, 15, 45, 135$. The differences grow ($10, 30, 90$) but each ratio is $3$, so the model is $y = 5 \cdot 3^x$.

A café finds that the line of best fit for daily hot-drink sales $y$ against the outdoor temperature $x$, in degrees, is $y = -2.5x + 140$. Predict sales on a $30$-degree day and interpret $-2.5$.

1. Predict: $y = -2.5(30) + 140 = -75 + 140 = 65$ drinks.
2. Interpret: for each $1$-degree increase in temperature, predicted sales decrease by $2.5$ drinks.

A line of best fit passes through $(4, 18)$ and $(10, 33)$. One data point is $(16, 44.5)$. What is the residual for this point?

1. Slope: $\frac{33 - 18}{10 - 4} = \frac{15}{6} = 2.5$.
2. Intercept: $18 = 2.5(4) + b$, so $b = 8$. The line is $y = 2.5x + 8$.
3. Predicted value at $x = 16$: $2.5(16) + 8 = 48$.
4. Residual: $44.5 - 48 = -3.5$.

Check: the line also gives $2.5(10) + 8 = 33$ at $x = 10$. The negative residual means the actual point lies $3.5$ units below the line.

• Residual backward. It is actual minus predicted. Predicted minus actual gives the opposite sign.
• Treating a prediction as certain. Answer choices with "exactly" or "will" are usually wrong; a model gives an estimate.
• Interpreting slope as a total. In $d = 3.2t + 7$, $3.2$ is the predicted growth per day, not the total size.
• Calling exponential data linear. Check differences and ratios. A constant ratio means exponential, even if the first few values look close to a line.
• Confusing association with causation. A strong scatterplot pattern does not show that $x$ causes $y$.