Least Squares Regression Line Math Example 3

Follow the full solution, then compare it with the other examples linked below.

Example 3

easy
Given y^=5+3x\hat{y} = 5 + 3x: (a) predict yy when x=4x=4, (b) interpret the slope, (c) does the line pass through the origin?

Solution

  1. 1
    (a) y^=5+3(4)=5+12=17\hat{y} = 5 + 3(4) = 5 + 12 = 17
  2. 2
    (b) Slope = 3: for each 1-unit increase in xx, yy increases by 3 units on average
  3. 3
    (c) At x=0x=0: y^=5โ‰ 0\hat{y} = 5 \neq 0; no, intercept = 5, not 0

Answer

(a) y^=17\hat{y}=17. (b) y increases 3 per unit of x. (c) No, line passes through (0,5).
The intercept (a=5) tells us the predicted y when x=0. The slope (b=3) is the rate of change. Only when a=0 does the line pass through the origin. These two parameters fully determine the linear model.

About Least Squares Regression Line

The unique straight line y^=a+bx\hat{y} = a + bx that minimizes the sum of squared vertical distances (residuals) between the observed data points and the line.

Learn more about Least Squares Regression Line โ†’

More Least Squares Regression Line Examples

Example 1 medium

Find the least-squares regression line for: [formula]: [formula]. Use [formula] and [formula].

Example 2 hard

The LSRL for predicting weight ([formula], kg) from height ([formula], cm) is [formula]. Interpret t

Example 4 hard

The LSRL has the property of minimizing [formula]. Explain why minimizing squared residuals (rather

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