Mathwords logoMathwords

y-hat — Definition, Formula & Examples

Y-hat (written ŷ) is the predicted value of the response variable that you get by plugging an x-value into a regression equation. It represents the point on the least-squares regression line corresponding to a given x.

In the context of a linear regression model y^=b0+b1x\hat{y} = b_0 + b_1 x, the quantity y^\hat{y} denotes the estimated value of the dependent variable yy for a specific value of the independent variable xx, as determined by the fitted regression equation.

Key Formula

y^=b0+b1x\hat{y} = b_0 + b_1 x
Where:
  • y^\hat{y} = Predicted value of the response variable
  • b0b_0 = y-intercept of the regression line
  • b1b_1 = Slope of the regression line
  • xx = Value of the explanatory variable

How It Works

After you calculate a least-squares regression line from your data, you can predict y-values for any given x by substituting into the equation. The result is ŷ, your best estimate of y at that x. The difference between an actual observed value yy and its predicted value y^\hat{y} is called a residual: e=y−y^e = y - \hat{y}. In AP Statistics, you will use ŷ whenever you interpret regression output, make predictions, or analyze residual plots.

Worked Example

Problem: A least-squares regression line for predicting test score from hours studied is ŷ = 50 + 8x. Predict the test score for a student who studies 5 hours.
Substitute: Replace x with 5 in the regression equation.
y^=50+8(5)\hat{y} = 50 + 8(5)
Calculate: Multiply and add to find the predicted score.
y^=50+40=90\hat{y} = 50 + 40 = 90
Answer: The predicted test score is ŷ = 90 points.

Why It Matters

In AP Statistics, nearly every regression problem asks you to find or interpret ŷ. Understanding ŷ is also essential for computing residuals and assessing how well a model fits, skills that carry directly into college-level statistics and data science work.

Common Mistakes

Mistake: Confusing ŷ (predicted value) with y (observed value).
Correction: The symbol ŷ always refers to the value the regression line predicts. The actual data point is y. The residual is their difference: e=y−y^e = y - \hat{y}.

Related Terms