Questions: Least-Squares Regression If the predicted value when (x=2) is (haty=31.9) and the actual value at (x=2) is measured to be (y=33.3), find the value of the residual at this point. residual =

Least-Squares Regression

If the predicted value when (x=2) is

(haty=31.9)

and the actual value at (x=2) is measured to be (y=33.3), find the value of the residual at this point.

residual =

Transcript text: Least-Squares Regression If the predicted value when $x=2$ is \[ \hat{y}=31.9 \] and the actual value at $x=2$ is measured to be $y=33.3$, find the value of the residual at this point. residual = $\square$ [?]

Solution

Solution Steps

Step 1: Understand the Concept of Residual

The residual in a regression analysis is the difference between the observed value and the predicted value. It is calculated as:

\[ \text{Residual} = y - \hat{y} \]

where $ y $ is the actual observed value and $ \hat{y} $ is the predicted value.

Step 2: Identify Given Values

From the problem, we have:

Predicted value at $ x = 2 $: $ \hat{y} = 31.9 $
Actual value at $ x = 2 $: $ y = 33.3 $

Step 3: Calculate the Residual

Using the formula for the residual:

\[ \text{Residual} = y - \hat{y} = 33.3 - 31.9 \]

Calculate the difference:

\[ \text{Residual} = 1.4 \]

Final Answer

The value of the residual at this point is:

\[ \boxed{1.4} \]

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