1. WARM UP: Use the Reciprocal Model to predict the y for an x = 81 20 2
11.1 3
7.7 4
5.9 6 11
2.2 16
1.5 x
1/y 1
.05 2
.09 3
.13 4
.17 6
.25 11
.45 16
.67
LinReg(x, 1/y)

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.
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.

3. Solve each equation for .
(Perform the Inverse Log function of raising both sides as the exponent of base 10.)
Use the Exponent property of xm+n = xm · xn
Use the Exponent property of
m Log x = Logxm

4. 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.
Reciprocal:
Exponential:
Power:

5. Examples B A
Minutes
Bacteria Population 2 15 3 34 4 77 5
173 6
389 7
876 8
1971
Minutes
Bacteria Population 2 33 3
104 4
226 5
418 6
690 7
1055 8
1521

6. A EXPONENTIAL FUNCTION
So, Take the LOG of only the y variable and perform a Regression
LinReg (L1=x , L3=LOG(y)). A
Minutes
Bacteria Population 2 15 3 34 4 77 5
173 6
389 7
876 8
1971
2.27
2.26
2.25

7. POWER FUNCTION
So, Take the LOG of Both the x and y variables and perform a Regression LinReg (L3=LOG(x) , L4=LOG(y)). B
Minutes
Bacteria Population 2 33 3
104 4
226 5
418 6
690 7
1055 8
1521
3.15
2.17
1.44

8. PAGE 238 #8,11,12

9. Use 2 digits for the year… 1980 = 80

11. Examine the Residuals of (x,y)
Examine the Residuals of (x,y). Then use the Reciprocal Model to find a Regression Line to model the data. Use your model to predict y when x = 10. x y 2 71 4 38 6 26 8 20 12 14 16 10
LinReg(L1, L2) L1 L2
L3 = 1/y 2 71
.01408 4 38
.02632 6 26
.03846 8 20
.05 12 14
.07143 16 10 .1
.125
LinReg(L1, L3)
R2 = 68.0
R2 = 99.9
Look at the Residuals
y-hat = 16.1 when x = 10

12. Another way to Analyze Non-Linear Data is to perform both
Examples B A
Minutes
Bacteria Population 2 15 3 34 4 77 5
173 6
389 7
876 8
1971
Minutes
Bacteria Population 2 33 3
104 4
226 5
418 6
690 7
1055 8
1521
Another way to Analyze Non-Linear Data is to perform both
Logarithmic Transformations on the Data and then compare the R2 and Residuals.

14. Plan B: Attack of the Logarithms

15. The Ladder of Powers -1 -1/2 x ½ 1 n Comment Model Name Power
Ratios of two quantities (e.g., mph) often benefit from a reciprocal. POWER MODEL
The reciprocal of the data -1
An uncommon re-expression, but sometimes useful.
Reciprocal square root
-1/2
Measurements that cannot be negative often benefit from a log re-expression.
Exponential x
Counts often benefit from a square root re-expression.
Square Root
Data with positive and negative values and no bounds are less likely to benefit from re-expression.
Linear (Raw Data) 1
Distributions that are skewed to the left.
Power (nth) n
Comment
Model Name
Power

16. Goals of Re-expression
Goal 1: Make the distribution of a variable (as seen in its histogram, for example) more symmetric.

17. Solve each equation for .
Warm – Up
Solve each equation for
(Flip both sides.)
(Perform the Inverse Log function of raising both sides as the exponent of base 10.)
Use the Exponent property of xm+n = xm · xn

18. Solve each equation for .
(Perform the Inverse Log function of raising both sides as the exponent of base 10.)
Use the Exponent property of xm+n = xm · xn
Use the Exponent property of
m Log x = Logxm

19. Chapter 10 Straightening Relationships
The relationship between fuel efficiency (in miles per gallon) and weight (in pounds) for late model cars looks fairly linear at first:

20. We can re-express fuel efficiency as gallons per hundred miles (a reciprocal) and eliminate the bend in the original scatterplot: