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Warm – Up
Round to 3 decimal places
•
•
•
•
•
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•
1.
2.
3.
4.
5.
6.
7.
0.699
Log(5) =
5
Log(105) =
LOG and 10x are
Inverse Functions
Log(5)
5
10
1.609
ln(5) =
148.413
e5 =
5
ln(e5) =
LN and e are
Inverse Functions
ln(5)
5
e =
Solving for
Solve each equation for
ŷ
1. yˆ = 1.82 + 0.4 x
x
2. yˆ = 1.82 ×(0.4)
4.
yˆ
1.82
yˆ = 3.82
ˆy = 3.822
0.4 x
.
when x = 5.
Linear Model:
Exponential Model:
3. yˆ = 1.82 x + 0.4 x + 3
2
ŷ
Power Model:
Power Model:
y = 14.5924
y = 3.82
y = 0.0186
y = 50.5
Solve each equation for
1
5.
yˆ
1.82
0.4 x
1
yˆ =
1.82 + .4 (5)
6. Log yˆ
ŷ
1.82
when x = 5.
Reciprocal Model:
(Flip both sides.)
0.4 x
Log yˆ = 1.82 + 0.4(5)
Log yˆ = 3.82
Log yˆ
3.82
ˆ
10
= y = 10
y = 6606.9345
y = 0.2618
Exponential Model:
(Perform the Inverse Log
function of raising both
sides as the exponent of
base 10.)
Solve each equation for
7. ln yˆ
1.82
0.4 x
ln yˆ = 1.82 + 0.4(5)
ln yˆ = 3.82
ln yˆ
3.82
e = yˆ = e
y = 45.604
ŷ
when x = 5.
Exponential Model:
(Perform the Inverse Natural
Log function of raising both
sides as the exponent of base
‘e’ .)
Chapter 10
Straightening Relationships
• We cannot use a Linear model unless the
relationship between the two variables is linear.
Often we can re-express or straighten bent
relationships so that we can fit and then use a
simple linear model.
• There are 3 ways to re-express data, taking:
Reciprocals, Roots, or Logarithms.
Choosing a Model - Part 1
x
y
1
25
2
18
3
14
5
10
10
6
15
4
20
3
Choosing a Model - Part 1
For Data that produces a curve and has a response variable
that is a ratio of two units, such as mpg, mph, GDP, use the
Reciprocal Model
x
yˆ
y
L1
L2
L3 = 1/y
1
LinReg(L1, L2)
25 R2 = 76.1
1
25
.04
2
18
2
18
.05556
3
14
3
14
.07143
5
10
5
10
.1
10
6
10
6
.16667
15
4
15
4
.25
20
3
20
3
.33333
Look at the
Residuals
0.0234
0.0152 x
1
1
ŷ
LinReg(L1, L3)
R2 = 99.8
LinReg(x, 1/y)
0.0234 0.0152x
Choosing a Model - Part 2
• For Exponential Models take the LOG(y) and then
perform a LinReg (x, LOG(y)). Use this model if there
exist a common ratio from successive y-values.
Log yˆ = a + bx
• For Power Models take the LOG(x) and LOG(y) and
then perform a LinReg (LOG(x), LOG(y)). Use this model
if no common ratio exist from successive y-values.
Log yˆ = a + b ×Log x
Choosing a Model
Reciprocal: – A ratio of two variables exists for y
Exponential or Power: -Perform a Stat, Calc
#0=ExpReg and a Stat, Calc #A=PwrReg.
Which ever model has the highest R2 will be
the model you choose.
PAGE 238 #1,2,3,10
PAGE 238 #1,2,3,10
Examples
A
Minutes
Bacteria
Population
2
3
4
5
6
7
8
15
34
77
173
389
876
1971
B
Minutes
Bacteria
Population
2
3
4
5
6
7
8
33
104
226
418
690
1055
1521
A
Minutes
Bacteria
Population
2
15
3
34
4
77
5
173
6
389
7
876
8
1971
EXPONENTIAL FUNCTION
2.27
2.26
So, Take the LOG of only the y variable and
perform a Regression
LinReg (L1=x , L3=LOG(y)).
Log (yˆ )= 0.4725 + 0.3529x
10
2.25
Log  yˆ 
yˆ  10
 100.47250.3529 x
0.4725183069
10
0.3529034054 x
yˆ = 2.9684 ×2.2537 x
B
Minutes
POWER FUNCTION
Bacteria
Population
2
33
3
104
4
226
5
418
6
690
7
1055
8
1521
3.15
2.17
So, Take the LOG of Both the x and y
variables and perform a Regression
LinReg (L3=LOG(x) , L4=LOG(y)).
Log (yˆ )= 0.6941+ 2.7568Log (x)
10
1.44
Log  yˆ 
0.69412.7568 Log  x 
 10
0.6940722249  2.756812639 Log x
yˆ  10
ˆy  100.6940722249 102.756812639Log x
0.6940722249
Log x 2.756812639
yˆ  10
10
ˆy = 4.9439 ×x 2.7568