Calculating the Least Squares Regression Line Lecture 49 Secs. 13.3.2 Wed, Dec 7, 2005

The Least Squares Regression Line The equation of the regression line is y^ = a + bx. Thus, we need to find the coefficients a and b. The formulas are or

Example Consider again the data set x y 2 3 5 9 6 12 16

Method 1 Compute the means and deviations for x and y. x y x –x y –y 2 3 -3 -6 5 -2 -4 9 6 12 1 16 4 7 x = 5 y = 9

Method 1 Compute the squared deviations, etc. x y 2 3 -3 -6 9 36 18 5 x –x y –y (x –x)2 (y –y)2 (x –x)(y –y) 2 3 -3 -6 9 36 18 5 -2 -4 4 16 8 6 12 1 7 49 28

Method 1 Find the sums of the last three columns. x y 2 3 -3 -6 9 36 x –x y –y (x –x)2 (y –y)2 (x –x)(y –y) 2 3 -3 -6 9 36 18 5 -2 -4 4 16 8 6 12 1 7 49 28 30 110 57

Method 1 Compute b: Then compute a:

Method 2 Consider again the data x y 2 3 5 9 6 12 16

Method 2 Compute x2, y2, and xy for each row. x y x2 y2 xy 2 3 4 9 6 5 25 15 81 45 12 36 144 72 16 256

Method 2 Then find the sums of x, y, x2, y2, and xy. x y x2 y2 xy 2 3 4 9 6 5 25 15 81 45 12 36 144 72 16 256 25 45 155 515 282

Method 2 Then find the sums of x, y, x2, y2, and xy. x y x2 y2 xy 2 3 4 9 6 5 25 15 81 45 12 36 144 72 16 256 x = 25 y = 45 x2 = 155 y2 = 515 xy = 282 25 45 155 515 282

Method 2 Compute b: Then compute a:

Example The second method is usually easier. By either method, we get the equation y^ = -0.5 + 1.9x.

TI-83 – Regression Line On the TI-83, we could use 2-Var Stats to get the basic summations. Then use the formulas for a and b. For our example, 2-Var Stats L1, L2 reports that n = 5 x = 25 x2 = 155 y = 45 y2 = 515 xy = 282

TI-83 – Regression Line Or we can use the LinReg function. Put the x values in L1 and the y values in L2. Select STAT > CALC > LinReg(a+bx). Press Enter. LinReg(a+bx) appears in the display. Enter L1, L2. Press Enter.

TI-83 – Regression Line The following appear in the display. The title LinReg. The equation y = a + bx. The value of a. The value of b. The value of r2 (to be discussed later). The value of r (to be discussed later).

Let’s Do It! Let’s Do It! 13.3, p. 819 – Oil Change Data. Use the TI-83. Enter the data into lists L1 and L2. Draw the scatterplot in order to decide whether a linear model is reasonable. Compute the regression line.