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Residuals

Residuals. Section 3.3. Recall from last week…. Regression lines are models for the overall pattern of a linear relationship between explanatory and response variables. Today we will look at…. Why are deviations also important?

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Residuals

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  1. Residuals Section 3.3

  2. Recall from last week… • Regression lines are models for the overall pattern of a linear relationship between explanatory and response variables.

  3. Today we will look at… • Why are deviations also important? • Because the LSRL is formed to minimize the distance vertical distance from the predictions to the observed values, they represent “left-over” variation. • These distances are called residuals.

  4. Residuals • The difference between an observed value of the response variable and the predicted value on the regression line. That is Residual = observed y – predicted y

  5. Example 3.14, p. 167

  6. Example 3.14, p. 167 • Plot scatterplot: 2nd, STAT PLOT • Plot LSRL: STAT, CALC, 8: LinReg(a + bx) L1, L2, Y1, Enter

  7. Example 3.14, p. 167

  8. Example 3.14, p. 167 LSRL: For child 1, who spoke at 15 months, we predict the score:

  9. Example 3.14, p. 167 The residual is Residual = observed y – predicted y = 95 – 92.97 = 2.03

  10. Why is this useful? • Because residuals show us how far the data fall from our regression line, examining them helps us to determine how well the line describes the data.

  11. Special Property of Residuals • The mean of the LS residuals are always zero!

  12. Residual Plot • Go to STAT PLOT • For Ylist, we will use RESID (found under 2nd STAT, 7: RESID) • Graph. ZoomStat (Zoom 9)

  13. Residual Plot

  14. Residual Plot • A scatterplot of the regression residuals against the explanatory variable. • Helps us assess the fit of a regression line.

  15. Residual Plots

  16. Residual Plots

  17. Residual Plots

  18. Influential Observations • An outlier is an observation that lies outside the overall pattern. • An observation is influential for a statistical calculation if removing it would markedly change the result of the calculation.

  19. Influential Observations

  20. Example 3.15, p. 172 • The strong influence of Child 18 makes the original regression of Gesell score on age at first word misleading. The original data have r2 = 0.41, which means the age a child begins to talk explains 41% of the variation on a later test of mental ability. This relationship is strong enough to be interesting to parents. If we leave out Child 18, r2 drops to only 11%. The apparent strength of the association was largely due to a single influential observation.

  21. Practice Problems • Exercises 3.46, 3.48

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