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Generating the Least Squares Regression Line

Generating the Least Squares Regression Line. Objective: To develop an understanding of the procedures used to generate a LSRL. Residuals. Residual = observed value – predicted value. residual = y – ŷ the distance of the point from the line

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Generating the Least Squares Regression Line

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  1. Generating the Least Squares Regression Line Objective: To develop an understanding of the procedures used to generate a LSRL.

  2. Residuals Residual = observed value – predicted value residual = y– ŷ the distance of the point from the line A least squares line is one that minimizes the sum of squared residuals

  3. Residuals continued… Some residuals will be an overestimate and others will be an underestimate Observed value Predicted value A residual is similar to the formula for deviation from chapter one. Knowing this, what do you think happens when you take the sum of all residuals?

  4. Determining LSRL • To determine the LSRL we will evaluate the vertical residuals. Here is the process: • Since some of the residuals will be positive and others will be negative, we will square them all. • Next we will add all the squared residuals (Σ) • This will be tested on all possible lines through the scatterplot and the line will the smallest sum of squared residuals will be used.

  5. Visualizing LSRL Least Squares Applet

  6. (3,10) y =.5(6) + 4 = 7 2 – 7 = -5 4.5 y =.5(0) + 4 = 4 0 – 4 = -4 -5 y =.5(3) + 4 = 5.5 10 – 5.5 = 4.5 -4 (6,2) (0,0) (0,0) Sum of the squares = 61.25

  7. (3,10) 6 Find y - y -3 (6,2) -3 (0,0) What is the sum of the deviations from the line? Will it always be zero? Use a calculator to find the line of best fit The line that minimizes the sum of the squares of the deviations from the line is the LSRL. Sum of the squares = 54

  8. The correlation coefficient and the LSRL are both non-resistantmeasures.

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