1. Calculating the Least Squares Regression Line
Lecture 45
Sec
Wed, Apr 25, 2007

2. 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

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

4. Method 1 Compute the means and deviations for x and y. x y x –x y –y2 3 -3 -6 5 -2 -4 9 6 12 1 16 4 7
x = 5
y = 9

5. 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

6. 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

7. Method 1
Compute b:
Then compute a:

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

9. Method 2 Compute x2, y2, and xy for each row. x y x2 y2 xy 2 3 4 9 6 525 15 81 45 12 36
144 72 16
256

10. Method 2 Then find the sums of x, y, x2, y2, and xy. x y x2 y2 xy 2 34 9 6 5 25 15 81 45 12 36
144 72 16
256 25 45
155
515
282

11. Method 2 Then find the sums of x, y, x2, y2, and xy. x y x2 y2 xy 2 34 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

12. Method 2
Compute b:
Then compute a:

13. Example The second method is usually easier.
By either method, we get the equation
y^ = x.

14. 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.

15. TI-83 – Regression Line 2-Var Stats L1, L2 reports that Etc., etc.
x = 25
x2 = 155
y = 45
y2 = 515
xy = 282
Etc., etc.

16. 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.

17. 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).

18. TI-83 – Regression Line
To graph the regression line along with the scatterplot,
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, Y1
Press Enter.

19. TI-83 – Regression Line Press Y= to see the equation.
Press ZOOM > ZoomStat to see the graph.

20. Example
Find the equation of the regression line for the Subway data on calories and cholesterol.
Calories (x)
350
290
330
320
370
280
310
230
Cholesterol (y) 50 20 45 15 35 25

21. Example 50 40 30
Cholesterol 20 10
Calories
200
250
300
350
400

22. Predicting y
How much cholesterol would we expect to find in a Subway sandwich that had 300 calories?

23. Predicting y
How much cholesterol would we expect to find in a Subway sandwich that had 300 calories?
Enter Y1(300).
Press Enter.
We get 25.6 calories.

24. Predicting y 50 40 30
Cholesterol 20 10
Calories
200
250
300
350
400

25. SST = Variation in the Data (y’s)50 40 30
Cholesterol 20 10
Calories
200
250
300
350
400

26. SSR = Variation in the Model50 40 30
Cholesterol 20 10
Calories
200
250
300
350
400

27. SSE = Residual Sum of Squares
It turns out that
SSE = SST – SSR.
That is,