1. 4.1: Linearizing Data

2. Logs of both sides: Linearized

3. Which model for prediction should you choose?
Plot the data and look for patterns.
Is there a linear pattern? Use Ch. 3.
If no to #1: try exponential model.
If no to #2, try power function model.
If no to #3, check for a dimensional relationship.
If no to #4, try the hierarchy of powers.

4. Example: Linearizing Curved Data “Playing” with the data in order to linearize it

5. Linearized data/Residuals

6. Final Model

7. #4.1, 4.2: Classwork
4.1 Hint
L1 = length, L2 = weight, L3 = cube-root of L2
4.2 Hint
L1=length, L2 = period, L3 = square root of L1

9. Example 4.8 (Transformed Exponential)

10. Steps to transform data
Transform (linearize) the data (exponential? Log the y’s; Power? Log both the x’s and the y’s)
Perform regression (check r, r-squared) and write new equation (with a and b)
Make residual plot (check that it’s a good model)
Perform an inverse transformation to get the model for the original data.

12. Ratio Test
To see if a set of data with a consistent increment in the x-values is growing exponentially, we calculate the ratios of y-values to previous y-values.
If these quotients are approximately the same, then the points are increasing (quotient >1) or decreasing (quotient <1) exponentially.

13. Example 4.5 Processor Years since 1970 Transistors
4004 1
2250
8008 2
2500
8080 4
5000
8086 8
29000
286 12
120000
386 15
275000
486DX 19
Pentium 23
Pent II 27
Pent III 29
Pent 4 30
Example 4.5
Gordon Moore predicted that the number of transistors on an integrated circuit chip would double every 18 months. Here’s the actual data:

14. Calculator Tip for Ratio Test
Copy the y values (in L2) into L3.
Since we want to calculate y2/y1, delete the first number in L3.
Since L2 and L3 need to be the same length, delete the last number in L2.
Define L4=L3/L2. These are the ratios you want.

15. If our data is growing exponentially and we plot the log of y against x, we should observe a straight line for the transformed data.

17. The table shows the federal debt (in trillions) for the years 1980 through 1991.
Construct a scatter plot. Perform an appropriate test to decide whether the data is exponential or not. Show the common ratios.
Calculate the logarithms of the y-values and extend the table above to show the transformed data Then perform least-squares regression on the transformed data. Write the LSRL equation for the transformed data.
What is the correlation coefficient?
Is this correlation between YEAR and FEDERAL DEBT? Explain briefly.
Now transform your linear equation back to obtain a model for the original federal debt data. Write the equation of this model.
Compare and comment on your models prediction for 1990 and 1991 to the actual federal debt.
Use your model to predict the national debt in the year 2000.
Would you use this model to predict future debts? Why/why not?
YEAR
DEBT
1980
.909
1981
.994
1982
1.1
1983
1.4
1984
1.6
1985
1.8
1986
2.1
1987
2.3
1988
2.6
1989
2.9
1990
3.2
1991
3.6