Math Problem Statement

Suppose you have the following data:

x 1 2 3 4 5 6
y 24 29 28 42 33 41

and the LSRL is ŷ =21.73+3.171x Find the residual value for x = 1.

Solution

To calculate the residual value for x=1x = 1, we use the formula for residuals:

Residual=yobservedypredicted\text{Residual} = y_{\text{observed}} - y_{\text{predicted}}

  1. Step 1: Find the observed value yobservedy_{\text{observed}} when x=1x = 1.
    From the given data, when x=1x = 1, yobserved=24y_{\text{observed}} = 24.

  2. Step 2: Find the predicted value ypredictedy_{\text{predicted}} using the least squares regression line (LSRL) equation:
    y^=21.73+3.171x\hat{y} = 21.73 + 3.171x Substitute x=1x = 1 into the equation:

y^=21.73+3.171(1)=21.73+3.171=24.901\hat{y} = 21.73 + 3.171(1) = 21.73 + 3.171 = 24.901

  1. Step 3: Calculate the residual:
    Residual=yobservedypredicted=2424.901=0.901\text{Residual} = y_{\text{observed}} - y_{\text{predicted}} = 24 - 24.901 = -0.901

Thus, the residual value for x=1x = 1 is -0.901.

Let me know if you'd like further clarification on any step!

Related Questions:

  1. How do you calculate the residual for other values of xx?
  2. What does a negative residual indicate about the fit of the regression line?
  3. How do you calculate the sum of squared residuals (SSR)?
  4. What is the significance of residual analysis in regression?
  5. How does the residual help in understanding the model's performance?

Tip:

Residuals help identify patterns that might indicate the model's weaknesses. If residuals show a non-random pattern, it may suggest that the linear regression model is not the best fit for the data.

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Math Problem Analysis

Mathematical Concepts

Linear Regression
Residuals
Least Squares Regression Line (LSRL)

Formulas

Residual = y_observed - y_predicted
ŷ = 21.73 + 3.171x

Theorems

Least Squares Regression Method

Suitable Grade Level

Grades 9-12