2.5 Correlation & Best-Fitting Lines
Algebra II
Mrs. Spitz
Fall 2008
Objectives
• Draw a scatter plot and find a prediction
equation.
• Solve problem using prediction equations
Assignment
• Pp. 103-104 #4-27
Information
• Line of best fit does not necessarily
contain any points from the data.
• When data is collected, the relation is
determined by the variables does not
usually form a straight line. However, it
may approximate a linear relationship.
• When this happens, a best-fit line can be
drawn and a prediction equation can be
determined using a process similar to that
used to determine an equation of a line
when you know two points.
Application
• Zimco Bottling Co. is promoting a continuing
education program for its employees. The
personnel director, Ms. Dirr would like to be able
to predict an employee’s salary if she knows the
number of year an employee attended college.
From the current personnel files, Ms. Dirr
randomly selected the files of ten employees.
She recorded each employee’s salary and
corresponding years of college for the
employee. See next slide.
Example 1:
Step 1: Graph the data points
• To determine the relationship between the number of
years of college and the salary, Ms. Dirr graphed the
data points to obtain a scatter plot. She found that the
points did not lie in a straight line, but clustered in linear
pattern. She draw a line suggested by this pattern of
points.
Years of college
3
2
4
6
2.5 7.5 7
1
5.5 4
Salary (in $1000)
15
20
22
47
19
10
30
18
32
28
Step 2: Select two points
• She then selected the
points (2.5, 19), (7,
32) on that line to
determine the
equation of the line.
To find the equation of
this line, she first
used the slope
formula and the two
points.
Step 3: Find the slope using the
two points
32  19 13
m

 2.9
7  2.5 4.5
Step 4: Let s represent an employee’s annual salary. Let c
represent the number of years of college education. Use
one of the points and the slope to find the prediction
equation.
s  2.9c  b
y = mx+b, where s = y,m = 2.9, and c = x.
32  2.9(7)  b
Point (7, 32) is used for values of c and s.
32  20.3  b
Simplify
11.7  b
Subtract 20.3 from both sides to isolate b.
Prediction Equation
• Prediction equation is
s = 2.9c + 11.7
• By using her prediction equation, Ms. Dirr
can encourage the employees with little
college education to go back to school.
For example, she can predict that with five
years of college education, their salary
might be $26,200.
Ex. 2: The table below shows the heights and the
corresponding ideal weights of adult women. Find a
prediction equation for this relationship.
• Step 1: Graph the data points.
Height (inches)
60
62
64
66
68
70
72
Weight (pounds)
105
111
123
130
139
149
158
• Draw a line that appears to be most
representative of the data. That’s your line
of best fit.
Step 2: Choose two points (62, 111) and (66, 130) from the
line to find the slope.
130  111 19
m

 4.8
66  62
4
Step 3: Now use the slope and one of the points in the
slope-intercept form to find the value of b.
y  mx  b
111  4.8(62)  b
111  297.6  b
 186.6  b
w  4.8h  186.6
Slope-intercept form
Substitute values into form.
Multiply 4.8 by 62 to simplify.
Subtract 297.6 from both sides.
Prediction equation
Is it accurate?
• The procedure for determining a prediction
equation is dependent upon your judgment. You
decide where to draw the best-fit line. You
decide which two points on the line are used to
find the slope and intercept. Your prediction
equation may be different from someone else’s.
The prediction equation is used when a rough
estimate is sufficient. For better analysis of the
data, statisticians normally use other, more
precise procedures, often relying on computers
and high-level programming.
Ex. 3: Draw a scatterplot and find two prediction equations to show
how typing speed and experience are related. Predict the typing speed
of a student who has 11 weeks of experience.
• Step 1: Graph the data points.
Experience (weeks)
4
7
8
1
Typing Speed (wpm)
33
45
46
20 40 30 38 22 52 44 42 55
6
3
5
2
9
6
7
• Draw a line that appears to be most
representative of the data. That’s your line
of best fit.
10
Step 2: Choose two points (5, 36) and (8, 49) from the line
to find the slope.
49  36 13
m
  4.3
85
3
Step 3: Now use the slope and one of the points in the
slope-intercept form to find the value of b.
y  mx  b
t  4.3e  b
36  4.3(5)  b
36  21.5  b
14.5  b
t  4.3e  14.5
Slope-intercept form
Let e stand for experience, t for typing speed.
Substitute values into form.
Multiply 4.3 by 5 to simplify.
Subtract 21.5 from both sides.
Prediction equation
Other
• One prediction equation is t = 4.3(e)+14.5.
• Another line can be suggested by using
different points. What ends up happening
is you are about 1/10 off, so either
prediction equation would produce a good
estimate.