Transcript Do Now - bYTEBoss

DO NOW

Find “r” on your calculator
X
Y
1
3
5
9
2
5
8
3
FINAL EXAM RECAP
Class Period
Average Test
Score
Highest Score
1st
135/200
(67%)
182.5/200
(91%)
4th
144/200
(72%)
178.5/200
(89%)
5th
135/200
(67%)
172/200
(86%)
MULTIPLE CHOICE
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
D
C
A
D
B
B
A
D
A
D
D
12. A
13. A
14. D
15. B
16. A
17. B
18. B
19. B
20. A
21. A
22. B
23. A
LEAST-SQUARES REGRESSION
January 22, 2013
OBJECTIVES
Find
the least-squares
regression line and use the line
to make predictions about data
Interpret the slope and the yintercept of the least-squares
regression line
VOCABULARY


Least-Squares Regression Line
 The line that minimizes the sum of the squared errors
(or residuals). It is the line that minimizes the sum of
the squared vertical distance between the observed
value of y and those predicted by the line
Residual
 The residual represents how close our prediction come
to the actual observation. The smaller the residual, the
better the prediction.
 The difference between the observed value of y and the
predicted value of y is the error.
FINDING AN EQUATION THAT DESCRIBES
LINEAR RELATED DATA
Once the scatter diagram and linear correlation
coefficient indicate that a linear relation exists
between two variables, we proceed to find a
linear equation that describes the relation
between the two variables.
 How do we go about finding this linear equation?


One way is to obtain a line that describes the relation
between two selected points from the data that
appear to provide a good fit and to find the equation
of the line through these points (Least-Squares
Regression Line)
BUT! BEFORE WE BEGIN…
Remember!!!
 When
working with least-square
regression lines, you will always
round out 4 decimal places!!
EXAMPLE PROBLEM #1
Club-Head Speed (x)
Distance (y)
(x, y)
100 mph
257 yards
(100, 257)
102
264
(102, 264)
103
274
(103, 274)
101
266
(101, 266)
105
277
(105, 277)
100
263
(100, 263)
99
258
(99, 258)
105
275
(105, 275)


Problem: The data in the table represents the
club-head speed and the distance a golf ball
travels for eight swings of the club. We
determined that these data are linearly related in
the previous chapter.
Directions:



(A). Find a linear equation that related club-head
speed and distance, by selecting two points and
finding the equation of the line containing the points
(B). Graph the line on the scatter diagram
(C). Use the equation to predict the distance a golf
ball will travel if the club-head speed is 104 miles per
hour.
APPROACH TO SOLVE (A)
1. Select two points so that a line drawn through
the points appears to give a good fit. Call the
points (x1, y1) and (x2, y2).
 2. Find the slope of the line containing the two
points using m = y2 – y1 / x2 – x1
 Use the point-slope formula, y – y1 = m(x – x1),
to find the equation of the line through the points
selected in step #1. Express the equation in the
form y = mx + b, where m is the slope and b is
the y-intercept

SOLUTION TO (A)
1. We select (x1, y1) = (99, 258) and (x2, y2) =
(105, 275), because a line drawn through these
two points seems to give a good fit.
 2. m = y2 – y1/ x2 – x1



275- 258 / 105- 99 = 17/6 = 2.8333
We use the point-slope formula to find the
equation of the line:





y – y1 = m(x –x1)
y – 258 = 2.8333(x – 99)
y – 258 = 2.8333x – 280.4967
y = 2.8333x – 22.4967
The slope of the line is 2.8333, and the yintercept is -22.4967
APPROACH AND SOLUTION TO (B)
For part (B), draw a line through the points
selected in Step #1 (99, 258) and (105, 275)
 See Graph (shown on Elmo)

APPROACH TO SOLVE (C)

For part (C), we let x = 104

Why 104? Because that is what is asked in the
problem. “Use the equation to predict the distance a
golf ball will travel if the club-head speed is 104 miles
per hour”
SOLUTION TO PART (C)
 We
let x = 104 in equation we found
in Part (A)


y = 2.8333x – 22.4967
y= 2.8333(104) – 22.4967

272.2 yards
Final Answer
We predict that a golf ball will travel
272.2 yards when it is hit with a club
head speed of 104 miles per hour.
LET’S PRACTICE 
X
3
4
5
7
8
Y
4
6
7
12
14
• Draw a scatter diagram treating x as the
explanatory variable and y as the response
variable
• Select two points from the scatter diagram
and find the equation of the line
containing the points selected
• (3,4) and (8,14)
• Graph the line found on the scatter
diagram
BUT IS LINE REALLY THE BEST?
For example, if the club-head speed is 103 miles
per hour, the predicted distance the ball will
travel is 2.8333(103) – 22.4967 = 269.3 yards.
 The observed distance for this club-head speed is
274 yards.
 Residual Formula

Observed y – predicted y
 = 274 – 269.3
 = 4.7 yards

It Looks Like We Have A Problem….
The Least-Squares
Regression Line To The
Rescue!
EXAMPLE PROBLEM
Find the least-squares regression line
 Predict the distance a golf ball will travel when
hit with a club-head speed of 103 miles per hour
 The observed distance the ball traveled when the
swing speed was 103 mph was 274 yards. Is this
distance above average or below average among
all golf balls hit with a swing speed of 103 mph?

IMPORTANT INFORMATION
r = 0.9386973
 Mean of x = 101.875
 Standard Deviation of x = 2.29518
 Mean of y = 266.75
 Standard Deviation of y = 7.74135
