Least-Squares Regression Section 3.3

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:

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

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.

Example 3.8, p. 150

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.

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.

Equation of LSRL

Facts about LSRL:

LSRL in the Calculator

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

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

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.

LSRL in the Calculator

To plot the line on the scatterplot by hand:

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

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

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!

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!

Practice Exercises Exercises 3.40, 3.41 p. 157