1. Section 4.1 Exponential Modeling
AP Statistics

2. Growth and Decay
Linear growth increases by a fixed amount in each equal time period
Exponential growth increase by a fixed percentage of the previous total.
AP Statistics, Section 4.1

3. Linear Growth X Y Difference 5 10-5=5 1 10 15-10=5 2 15 20-15=5 3 205
10-5=5 1 10
15-10=5 2 15
20-15=5 3 20
25-20=5 4 25
30-25=5 30
35-30=5 6 35
40-35=5 7 40
AP Statistics, Section 4.1

4. Linear Growth X Y Difference 5 10-5=5 1 10 25-10=15 4 25 40-25=15 7 405
10-5=5 1 10
25-10=15 4 25
40-25=15 7 40
55-40=15 55
75-55=20 14 75
95-75=20 18 95
115-95=20 22
115
AP Statistics, Section 4.1

5. Exponential Growth
is interesting, but what we really want is a linear model
AP Statistics, Section 4.1

6. Exponential Growth X Y Difference 1 2-1=1 2 4-2=2 4 8-4=4 3 8 16-8=81
2-1=1 2
4-2=2 4
8-4=4 3 8
16-8=8 16
32-16=16 5 32
64-32=32 6 64
128-64=64 7
128
AP Statistics, Section 4.1

7. Exponential Growth X Y Ratio 1 2/1=2 2 4/2=2 4 8/4=2 3 8 16/8=2 161
2/1=2 2
4/2=2 4
8/4=2 3 8
16/8=2 16
32/16=2 5 32
64/32=2 6 64
128/64=2 7
128
AP Statistics, Section 4.1

8. Exponential Growth X Y Ratio 1 2/1=2 2 8/2=4 3 8 32/8=4 5 32 128/32=41
2/1=2 2
8/2=4 3 8
32/8=4 5 32
128/32=4 7
128
512/128=4 9
512
4096/512=8 12
4096
32768/4096=8 15
32768
AP Statistics, Section 4.1

9. AP Statistics, Section 4.1

10. Year Subscribers Ratios Log(Sub) 1990 5,283 3.722880611 1993 16,009
1994
24,134
1995
33,786
1996
44,043
1997
55,312
1998
69,209
1999
86,047
AP Statistics, Section 4.1

11. AP Statistics, Section 4.1

12. Year Subscribers Ratios Log(Sub) 1990 5,283 3.722880611 1993 16,009
1994
24,134
1995
33,786
1996
44,043
1997
55,312
1998
69,209
1999
86,047
AP Statistics, Section 4.1

13. AP Statistics, Section 4.1

14. AP Statistics, Section 4.1

15. Summary We typically start by putting X data in L1 and Y data in L2.
If data is in years, then declare the first year to be “year 0” and let the other years be coded as the run of years after “year 0”
Run the linear regression “LinReg (a+bx)”
AP Statistics, Section 4.1

16. Summary
After doing linear regression X vs. Y, we see evidence that the set is not linear
The scatterplot look exponential
A curved pattern in the residual plot
AP Statistics, Section 4.1

17. Summary Suspecting that X and Y have an exponential relationship…
Transform the Y data. Put log(Y) into L3.
Confirm the exponential relationship…
By seeing the correlation is stronger for X vs log(Y) than it is for X vs Y
By seeing to pattern in the residual plot of X vs log(Y)
AP Statistics, Section 4.1

18. Summary After confirming the exponential relationship between X and Y
Transform the linear relationship
Log(y-hat) = a + bx
Into the exponential relationship
y-hat = 10a *(10b)x
AP Statistics, Section 4.1

19. Assignment Exercises: 4.6, 4.8, 4.9, 4.11, 4.19, 4.21
AP Statistics, Section 4.1