Download Linear Regression

Linear Regression
Section 3.3
Warm Up
• In many communities there is a strong
positive correlation between the amount of
ice cream sold in a given month and the
number of drownings that occur in that
month. Does this mean that ice cream
causes drowning? If not, can you think of
other alternatives for the strong
association?
Warm Up #2……
• Explain why one would expect to find
a positive correlation between the
number of fire engines that respond
to a fire and the amount of damage
done in the fire.
Regression Line……
• If the value of the
correlation
coefficient is
significant, the
next step is to find
the equation of the
regression line.
• Regression Line –
The data’s line of
best fit which is
determined by the
slope and yintercept.
Regression Analysis……
• It finds the
equation of the line
that best
describes the
relationship
between the 2
variables.
• Primary Purpose:
To Make
Predictions
*This is a test
question.
Prediction Models……
a. Linear : y  ax  b
b. Quadratic : y  ax 2  bx  c
c. Exponential : y  a(b x )
d . Logarithmic : y  a log b x
Remember Algebra?......
• The slope
• Slope:
intercept form of a
The change in y
line was y = mx + b
over the change in
where m is the
x.
slope and b is the
• Y-intercept:
y-intercept
where the line
crosses the y-axis.
Line of Best Fit……
• The equation used
to find the line of
best fit is
y = ax + b
• where
“a” = slope
and
“b” = y-intercept
Computational
Formulas……y = ax + b
• To find a:
( x  x)( y  y )
a
2
( x  x )
• To find b:
b  y  (a  x)
Example 1……
• Find the equation
of the line of best
fit.
• Predict the # of
sales when 5 ads
are sold.
# of ads
# of sales
3
7
4
6
2
5
6
10
4
8
Go by the formula……These
are the lists you will need.
x
y
xx
y y
( x  x)( y  y)
( x  x) 2
First……
• Find the mean of x
and the mean of y
and write it down.
• Put x’s in L1 – stat
calc one var stats
L1
• Put y’s in L2 – stat
calc one var stats
L2
Means of x and y……
Let’s fill in the lists……
L1
L2
L3 = L1 - 3.8
L4 = L2 - 7.2
L5 =L3 x L4
L6 = L3 squared
x
y
x - xbar
y - ybar
(x-xbar)(y-ybar)
(x - xbar) squared
3
7
-0.8
-0.2
0.16
0.64
4
6
0.2
-1.2
-0.24
0.04
2
5
-1.8
-2.2
3.96
3.24
6
10
2.2
2.8
6.16
4.84
4
8
0.2
0.8
0.16
0.04
10.2
8.8
Compute “a”……
10.2
a
 1.159090909  1.16
8.8
Compute “b”……
b  7.2  (1.159090909)(3.8)  2.8
Plug into y = ax + b……
• Answer:
y = 1.16x + 2.8
Predict ……
• Predict the number of sales when 5
ads are sold.
Y = 1.16(5) + 2.8 = 8.6 = 9 sales
Example 2……
• A. Find the equation
of the line of best fit.
• B. Predict hours of
exercise if the person
is 35 yrs old.
• C. Predict the age if
they exercise 9 hours
per week.
Age
Exercise
18
10
26
5
32
2
38
3
52
1.5
59
1
Find the means……
• X-Values:
• Y-Values:
The lists……
L1
L2
L3 = L1 - 37.5
L4 = L2 - 3.75
L5 =L3 x L4
L6 = L3 squared
x
y
x - xbar
y - ybar
(x-xbar)(y-ybar)
(x - xbar) squared
18
10
-19.5
6.25
-121.9
380.25
26
5
-11.5
1.25
-14.38
132.25
32
2
-5.5
-1.75
9.625
30.25
38
3
0.5
-0.75
-0.375
0.25
52
1.5
14.5
-2.25
-32.63
210.25
59
1
21.5
-2.75
-59.13
462.25
-218.75
1215.5
Compute “a” and “b”……
 218.75
a
 .18
1215.5
b  3.75  (.18)(37.5)  10.50
Equation: y = mx + b
• Plug into the formula for the
equation of the trend line.
Y = -.18x + 10.50
Predictions……
• Find y when x = 35.
• Find x when y = 9.
• Y = -.18(35) + 10.50
•
•
•
•
•
• Y = 4.2 hours
9 = -.18x + 10.50
9-10.50 = -.18x
-1.5 = -.18x
X = 8.3
X = 8 years