Represent two quantitative variables on a scatter plot, describe the relationship, fit a linear model, and interpret its slope and intercept in context (TN A1.S.ID.C.6, A1.S.ID.C.7).

What this topic is asking

Standards A1.S.ID.C.6 and A1.S.ID.C.7 move to two quantitative variables. You plot the pairs on a scatter plot, describe the relationship (direction, form, strength), fit a linear model (line of best fit), and interpret its slope and intercept in context. The graded skills are describing association and reading the model as a rate and a starting value.

Describing the association

From a scatter plot, report three things:

• Direction: positive (points trend up to the right) or negative (down to the right).
• Form: roughly linear, curved, or no pattern.
• Strength: strong (tight cluster) or weak (loose scatter).

For example, hours studied versus test score usually shows a positive, roughly linear, moderately strong association.

The linear model: slope and intercept

When the pattern is roughly linear, a line of best fit $y = mx + b$ summarizes it. In context:

• Slope $m$: the predicted change in $y$ for each one-unit increase in $x$, a rate. "$6$ points per hour studied."
• $y$-intercept $b$: the predicted $y$ when $x = 0$. "A predicted score of $52$ with no studying."

How TNReady examines this topic

• Multiple choice. Describe the association, or interpret the slope or intercept of a model.
• Numeric response. Predict a value from the fitted equation.
• Graphing. Identify or draw a reasonable line of best fit.

A clarifying idea is that the line of best fit is an ordinary linear function: its slope is read the same way as any slope (rate of change), and the model is used to predict by substituting, exactly like evaluating a function.

Why interpretation and prediction range matter

Two ideas elevate an answer on this topic. First, the slope and intercept must be stated in context with units: "$8$ ice creams per degree" and "predicted sales of $-100$ at $0$ degrees," not just "$8$" and "$-100$." The units come from the axes, and the interpretation is what earns credit, the same lesson as interpreting key features. Second, predictions are trustworthy mainly within the range of the data (interpolation). Pushing the model far beyond the observed $x$-values (extrapolation) can give nonsense, as the $-100$ ice creams at $0$ degrees shows: no real data near $0$ degrees supported that part of the line. A good habit is to check whether a prediction's $x$-value lies inside the data you were given, and to flag a meaningless intercept rather than report it as fact. This is also where the next topic comes in: a strong-looking line does not by itself prove that $x$ causes $y$.

Try this

Q1. A model is $y = 3x + 10$ for years of experience ($x$) versus salary in thousands ($y$). What does the slope mean? [1 point]

• Cue. Each additional year of experience is associated with about \3000$ more salary.

Q2. Using $y = 3x + 10$, predict the salary for $8$ years of experience. [1 point]

• Cue. $y = 3(8) + 10 = 34$, i.e. \34{,}000$.