Represent two-variable data on a scatter plot, fit a linear model (line of best fit), and interpret the slope and intercept in context, using the model to predict (Ohio S-ID.6, S-ID.7).

What this topic is asking

Ohio standards S-ID.6 and S-ID.7 ask you to plot two-variable data on a scatter plot, fit a linear model (line of best fit), and interpret its slope and intercept in context, then use it to predict. This connects statistics to lines, the same $y = mx + b$ now describes a trend in data. It is a reliable Statistics-category skill, usually on the calculator part.

Reading a scatter plot

A scatter plot shows the relationship between two numerical variables.

A clear up-right or down-right pattern signals a linear model is reasonable.

Fitting and interpreting the line

The line of best fit summarizes the trend; its slope and intercept carry meaning.

Predicting and its limits

Substitute an $x$-value into $\hat{y} = mx + b$ to predict a $y$. Predicting within the range of the data (interpolation) is reliable; predicting far outside it (extrapolation) is risky, the trend may not continue, so a model can give a nonsensical prediction beyond the data.

How Ohio examines this topic

• Equation response. Interpret a slope or intercept, or predict a value from the line of best fit.
• Multiple choice and multiple-select. Identify the direction of association, or match a scatter plot to a model.
• Graphing. Plot points or sketch a reasonable trend line.

Why slope and intercept mean the same as for any line

A line of best fit is still $y = mx + b$, so its slope and intercept carry the same meanings as in any linear model, now applied to data. The slope is rise over run, the change in the response $y$ for a one-unit change in the predictor $x$, which in context is a rate: dollars per degree, points per study hour, centimeters per year. The intercept is the value of $y$ when $x = 0$, the baseline prediction. The only new wrinkle is the hat on $\hat{y}$: it signals the line gives a predicted (estimated) value, not the exact data, because real points scatter around the line rather than lying on it. Carrying over your line-knowledge from algebra, and adding "this is a prediction," is what makes interpreting a line of best fit straightforward.

Why extrapolation is risky

Predicting with the line of best fit is trustworthy within the range of the observed data, because the line was fit to that range and the pattern is known to hold there. Extrapolating far beyond the data assumes the same linear trend continues, an assumption the data cannot support. A study-time model might predict ever-higher scores for absurd numbers of hours, past $100\%$ or beyond what is possible, and a temperature-sales model would predict negative sales at very low temperatures. The relationship may bend, flatten, or break down outside the observed window. This is why interpreting an intercept can be misleading when $x = 0$ lies far from the data, and why the test rewards recognizing that a prediction well outside the data range is unreliable, even when the arithmetic is correct.

Try this

Q1. A line of best fit is $\hat{y} = -3x + 40$. Interpret the slope. [1 point]

• Cue. $y$ decreases by about $3$ per one-unit increase in $x$ (negative association).

Q2. For $\hat{y} = 2x + 10$, predict $y$ when $x = 7$. [1 point]

• Cue. $\hat{y} = 2(7) + 10 = 24$.