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Least-Squares Regression

Least-Squares Regression. Section 3.3. Recall from 3.2:. Correlation measures the strength and direction of a linear relationship between two variables. How do we summarize the overall pattern of a linear relationship? Draw a line!. Least-Squares Regression.

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Least-Squares Regression

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  1. Least-Squares Regression Section 3.3

  2. Recall from 3.2: Correlation measures the strength and direction of a linear relationship between two variables. How do we summarize the overall pattern of a linear relationship? Draw a line!

  3. Least-Squares Regression A method for finding a line that summarizes the relationship between two variables, but only in a specific setting.

  4. Regression Line “Best-fit Line” A straight line that descirbes how a response variable y changes as an explanatory variable x changes. Predict y from x. Requires that we have an explanatory variable and a response variable.

  5. Example 3.8, p. 150

  6. Least-Squares Regression Line Because different people will draw different lines by eye on a scatterplot, we need a way to minimize the vertical distances.

  7. Least-Squares Regression Line The LSRL of y on x is the line that makes the sum of the squares of the vertical distances of the data from the line as small as possible.

  8. Equation of LSRL For data with explanatory variable x and response variable y for n individuals, find the means and standard deviations of each variable and their correlation. The least squares regression is the line

  9. Equation of LSRL With slope: And intercept:

  10. Facts about LSRL: Slope: - The change of one std. dev. in x results in a change of r std. dev. in y. 2. x & y assignments matter. 3. LSRL will always go through

  11. LSRL in the Calculator

  12. LSRL in the Calculator After you’ve entered data, STAT PLOT. ZoomStat (Zoom 9)

  13. LSRL in the Calculator To determine LSRL: Press STAT, CALC, 8:LinReg(a+bx), Enter

  14. LSRL in the Calculator To get the line to graph in your calculator: Press STAT, CALC 8:LinReg(a+bx) L1, L2, Y1 Now look in Y =. Then look at your graph.

  15. LSRL in the Calculator

  16. To plot the line on the scatterplot by hand: Use the equation for for two values of x, one near each end of the range of x in data. Plot each point.

  17. For Example: Use the equation: . Smallest x = 0, Largest x = 52 Use these two x-values to predict y.

  18. For Example: Use the equation: . (0, 1.0892), (52, 10.9172) To get another point, use STAT, CALC, 2:2-Var Stats. We can use .

  19. Extrapolation Suppose that we have data on a child’s growth between 3 and 8 years of age. The least-squares regression line gives us the equation , where x represents the age of the child in years, and will be the predicted height in inches. What if you wanted to predict the height of a 25 year old girl? Would this equation be appropriate to use? NO!

  20. Extrapolation Extrapolation is the use of a regression line for prediction far outside the domain of values of the explanatory variable x that you used to obtain the line or curve. Such predictions are often not accurate. That’s over 7’ 9” tall!

  21. Practice Exercises Exercises 3.40, 3.41 p. 157

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