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Least-Squares Regression Section 3.3

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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! Recall from 3.2:

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

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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.

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Example 3.8, p. 150

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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.

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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.

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Equation of LSRL

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Facts about LSRL:

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LSRL in the Calculator

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After you’ve entered data, STAT PLOT. ZoomStat (Zoom 9)

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

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LSRL in the Calculator To get the line to graph in your calculator: Press STAT, CALC 8:LinReg(a+bx) L 1, L 2, Y 1 Now look in Y =. Then look at your graph.

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LSRL in the Calculator

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To plot the line on the scatterplot by hand:

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

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For Example: (0, 1.0892), (52, 10.9172)

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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!

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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!

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Practice Exercises Exercises 3.40, 3.41 p. 157

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