Section 4 1 Exponential Modeling AP Statistics www

Growth and Decay Linear growth increases by a fixed amount in each equal time

Linear Growth X Y Difference 0 5 10 -5=5 1 10 15 -10=5 2

Linear Growth X Y Difference 0 5 10 -5=5 1 10 25 -10=15 4

Exponential Growth n is interesting, but what we really want is a linear model

Exponential Growth X Y Difference 0 1 2 -1=1 1 2 4 -2=2 2

Exponential Growth X Y Ratio 0 1 2/1=2 1 2 4/2=2 2 4 8/4=2

Exponential Growth X Y Ratio 0 1 2/1=2 1 2 8/2=4 3 8 32/8=4

Year Subscribers Ratios Log(Sub) 1990 5, 283 3. 722880611 1993 16, 009 4. 204364205

Summary We typically start by putting X data in L 1 and Y data

Summary n After doing linear regression X vs. Y, we see evidence that the

Summary n Suspecting that X and Y have an exponential relationship… ¨ Transform n

Summary n After confirming the exponential relationship between X and Y ¨ Transform n

Assignment n Exercises: 4. 6, 4. 8, 4. 9, 4. 11, 4. 19, 4.

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Section 4. 1 Exponential Modeling AP Statistics www. toddfadoir. com/apstats AP Statistics, Section 4. 1

Growth and Decay Linear growth increases by a fixed amount in each equal time period n Exponential growth increase by a fixed percentage of the previous total. n AP Statistics, Section 4. 1 2

Linear Growth X Y Difference 0 5 10 -5=5 1 10 15 -10=5 2 15 20 -15=5 3 20 25 -20=5 4 25 30 -25=5 5 30 35 -30=5 6 35 40 -35=5 7 40 AP Statistics, Section 4. 1 3

Linear Growth X Y Difference 0 5 10 -5=5 1 10 25 -10=15 4 25 40 -25=15 7 40 55 -40=15 10 55 75 -55=20 14 75 95 -75=20 18 95 115 -95=20 22 115 AP Statistics, Section 4. 1 4

Exponential Growth n is interesting, but what we really want is a linear model AP Statistics, Section 4. 1 5

Exponential Growth X Y Difference 0 1 2 -1=1 1 2 4 -2=2 2 4 8 -4=4 3 8 16 -8=8 4 16 32 -16=16 5 32 64 -32=32 6 64 128 -64=64 7 128 AP Statistics, Section 4. 1 6

Exponential Growth X Y Ratio 0 1 2/1=2 1 2 4/2=2 2 4 8/4=2 3 8 16/8=2 4 16 32/16=2 5 32 64/32=2 6 64 128/64=2 7 128 AP Statistics, Section 4. 1 7

Exponential Growth X Y Ratio 0 1 2/1=2 1 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 8

AP Statistics, Section 4. 1 9

Year Subscribers Ratios Log(Sub) 1990 5, 283 3. 722880611 1993 16, 009 4. 204364205 1994 24, 134 1. 507527 4. 382629308 1995 33, 786 1. 399934 4. 528736778 1996 44, 043 1. 303587 4. 643876893 1997 55, 312 1. 255864 4. 742819362 1998 69, 209 1. 251247 4. 840162574 1999 86, 047 1. 243292 4. 934735733 AP Statistics, Section 4. 1 10

AP Statistics, Section 4. 1 11

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Summary We typically start by putting X data in L 1 and Y data in L 2. n 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” n Run the linear regression “Lin. Reg (a+bx)” n AP Statistics, Section 4. 1 15

Summary n 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 16

Summary n Suspecting that X and Y have an exponential relationship… ¨ Transform n the Y data. Put log(Y) into L 3. 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 17

Summary n After confirming the exponential relationship between X and Y ¨ Transform n Log(y-hat) = a + bx ¨ Into n the linear relationship the exponential relationship y-hat = 10 a *(10 b)x AP Statistics, Section 4. 1 18

Assignment n Exercises: 4. 6, 4. 8, 4. 9, 4. 11, 4. 19, 4. 21 AP Statistics, Section 4. 1 19