Prediction Math Example 3

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

Example 3

easy
Using the model y^=3x−5\hat{y} = 3x - 5, predict yy when x=4x = 4 and x=0x = 0. Then find xx when y^=25\hat{y} = 25.

Solution

  1. 1
    x=4x=4: y^=3(4)−5=12−5=7\hat{y} = 3(4) - 5 = 12 - 5 = 7
  2. 2
    x=0x=0: y^=3(0)−5=−5\hat{y} = 3(0) - 5 = -5 (y-intercept)
  3. 3
    y^=25\hat{y}=25: 25=3x−5⇒3x=30⇒x=1025 = 3x - 5 \Rightarrow 3x = 30 \Rightarrow x = 10

Answer

y^(4)=7\hat{y}(4)=7; y^(0)=−5\hat{y}(0)=-5; x=10x=10 when y^=25\hat{y}=25.
Linear models allow prediction in both directions: given x, find predicted y; or given desired y, solve for x. The y-intercept (x=0) is the baseline prediction when the explanatory variable equals zero.

About Prediction

A prediction is a model-based estimate of an unknown or future value, accompanied by a measure of confidence or uncertainty.

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