Fit and interpret a linear model for bivariate data, use the line of best fit to predict and to interpret slope and intercept in context, and distinguish correlation from causation.

What this topic is asking

The Statistics and Probability category requires fitting and interpreting a linear model for two-variable data (the S-ID standards). On the Grade 10 MCAS you use a line of best fit to make predictions, interpret its slope and intercept in context, judge correlation, and recognize that correlation is not causation. The interpretation of slope and intercept, and the caution about extrapolation and cause, are the points the test most rewards.

The line of best fit

A line of best fit summarizes the linear trend in two-variable data with a single equation $y = mx + b$. It is the line that comes closest to the points overall, and the MCAS gives you the equation rather than asking you to compute it by hand. Once you have it, the same reading skills as for any line apply:

• The slope $m$ is the predicted change in $y$ for each one-unit increase in $x$. In context it is a rate: points per study hour, dollars per year.
• The intercept $b$ is the predicted value of $y$ when $x = 0$, the starting value of the trend.

For $y = 2.5x + 12$ (study hours to score), the slope says each extra hour predicts about 2.5 more points, and the intercept says a student who does not study is predicted to score 12.

Making predictions

To predict, substitute an $x$-value into the line of best fit. For $y = 2.5x + 12$ at $x = 6$: $y = 2.5(6) + 12 = 27$. This is a predicted value, the trend's estimate, not a guaranteed outcome.

Two cautions the MCAS rewards:

• A prediction is an estimate around which individual data points scatter; the line does not fit every point exactly.
• Predicting far outside the range of the data (extrapolation) is unreliable, because the linear trend may not continue beyond the observed values.

Correlation

The correlation coefficient $r$ measures the strength and direction of a linear relationship, on a scale from $-1$ to $1$:

• $r$ near $+1$: strong positive linear relationship (points rise tightly).
• $r$ near $-1$: strong negative linear relationship (points fall tightly).
• $r$ near $0$: little or no linear relationship.

So $r = 0.92$ is a strong positive correlation, and $r = -0.15$ is a weak negative one. The sign of $r$ always matches the slope's sign.

Correlation is not causation

The most important reasoning point: a strong correlation between two variables does not prove that one causes the other. Both might be driven by a third factor (a lurking variable), or the link could be coincidence. Ice cream sales and drowning rates correlate, but ice cream does not cause drowning; hot weather drives both. The MCAS tests whether you can describe a relationship as an association while resisting an unjustified causal claim.

Try this

Q1. For $y = -4x + 50$, what does the slope $-4$ mean in context?

• Cue. $y$ decreases by 4 for each one-unit increase in $x$.

Q2. A correlation of $r = 0.05$ describes what kind of linear relationship?

• Cue. Very weak (almost none).