Linear Regression Models

A line through the cloud

• When a scatterplot looks linear, summarize it with a regression line 回归线.
• We write the predicted response as $\hat{y} = a + bx$ (the hat means "predicted").
• $a$ is the $y$-intercept, $b$ is the slope — the line's recipe.
• The line lets us predict $y$ from any $x$ and describe the trend precisely.

Reading the slope

• The slope $b$ is the predicted change in $y$ for each $1$-unit increase in $x$.
• In context: "each extra hour of study predicts about $b$ more points."
• Its sign matches the direction of the association (positive or negative).
• The slope is where almost all the interpretation credit lives on the exam.

Reading ŷ = a + bx: the slope is the predicted change per unit of x; the intercept is the prediction at x = 0

Reading the intercept

• The intercept $a$ is the predicted $y$ when $x = 0$.
• Sometimes meaningful ("baseline score with zero study"), sometimes not.
• If $x = 0$ is far outside the data, the intercept is just a mathematical anchor.
• Interpret it in context — and be honest when $x=0$ makes no real sense.

Don't extrapolate

• Use the line only within the range of the observed $x$ values.
• Extrapolation 外推 — predicting far beyond the data — is unreliable.
• The linear pattern may simply not continue out there.
• A model that fits students studying $0$–$8$ hours says nothing trustworthy about $40$ hours.

Write from a prediction of 19

• A line is predicted y equals 4 plus 3 times x.
• x is 5. The lesson intercept context uses 67.