FITTING EXPONENTIAL MODELS TO DATA Supplement to Unit 9C MATH Q(t) = Q 0 (1 + r) t. Q(t) = Q 0 a t,

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1 FITTING EXPONENTIAL MODELS TO DATA Supplemen o Un 9C MATH 01 In he handou we wll learn how o fnd an exponenal model for daa ha s gven and use o make predcons. We wll also revew how o calculae he SSE and average error. We wll learn how o fnd he exponenal model ha bes fs a se of gven daa. An Alernae Form of he Exponenal Funcon The exponenal funcon used n he ex s Ths formula s equvalen o where a = 1 + r. Q() = Q 0 (1 + r). Q() = Q 0 a, Fndng an Exponenal Model for Daa and Makng Predcons Frs, we consder he followng able ha gves he populaon for San Dego, Calforna from 1960 o Year Pop. (housand) Change The hrd column of hs able shows (for each decade year) he change n populaon durng he precedng decade. We see he change n he populaon of San Dego ncreased each decade. We mgh wonder wheher hs qualfes as almos exponenal populaon growh. So, we wll plo he daa and look. We plo he year on he x-axs and he populaon on he y-axs. Fgure 1

2 2 Fng Exponenal Models o Daa We can see ha daa graphed n Fgure 1 has an upward bend o. Ths ndcaes ha an exponenal model may f he daa well. Bu, how can we fnd an exponenal funcon ha passes hrough or near each daa pon? One way s o smply pass an exponenal funcon hrough he frs and he las daa pons. To make hs easer, we wll le be he number of years afer Thus, our daa wll look lke: (years snce 1960) P (n housand) We wll use an exponenal model of he form P() = P 0 a. We see ha he nal populaon (he populaon when = 0) s 573 housand. So, P 0 = 573. When =, we have P() = Thus, we need o solve he equaon 1111 = 573 a 1111 = a = a = a a Thus, an exponenal model for he daa s P() = 573. NOTE: Snce = , he annual growh rae s 2.23%. Fgure 2 below shows he daa and he exponenal modeled graphed ogeher. We can see ha he model fs he daa very closely. Fgure 2

3 3 Fng Exponenal Models o Daa Now, le s use our model o predc he populaon of San Dego n 1985 and n The year 1985 s 25 years afer So, o predc he populaon n 1985, we subsue = 25 no our model. P(25) = Thus, he populaon of San Dego n 1985 was approxmaely housand people. The year 2000 s 40 years afer So, o predc he populaon n 2000, we subsue = 40 no our model. P(40) = Thus, he populaon of San Dego n 2000 was approxmaely housand people, or mllon people. We can also use he model o predc he year durng whch he populaon reaches a ceran number. For example, we could use our model above o predc when he populaon of San Dego wll reach 1500 housand people. To do hs, we se P() = 1500 and solve for. We wll need o use arhms o do hs = = = = = Thus, he populaon wll reach 1500 housand n years. To deermne he calendar year, we add 1960 and 43 o ge he year To deermne he monh we mulply 0.63 by 12 o ge Hence he populaon wll reached 1500 housand n Augus Measurng How Closely he Model Fs he Daa Now we wll fnd he SSE and average error for he model we found above, whch s P() = 573.

4 4 Fng Exponenal Models o Daa P P() Error, E 2 E (Acual) (Predced) P P() So, he SSE = = And he average error s average error = Fndng he Bes-F Exponenal Model for Gven Daa The bg queson s: how do we fnd he bes-f exponenal model for he daa. In shor, we fnd by fndng he exponenal model ha makes boh he SSE and average error as small as possble. Our calculaors wll fnd he bes-f exponenal model for us. The seps are oulned below. 1. Selec STAT, 1:Ed. 2. Ener he x-values for he daa n L1 and he y-values n L2. 3. Press 2nd, MODE o reurn o he home screen. 4. Press STAT and arrow over o CALC. 5. Selec 0:ExpReg. 6. Then ener L1 and L2 (or whch ever lss you have your x- and y-values sored n). 7. Press L1,L2. 8. Press ENTER. Example: (a) Fnd he bes-f exponenal model for he populaon daa for San Dego, Calforna. (b) Use your model o predc he populaon n 1985 and (c) Fnd he year n whch he populaon wll reach 1500 housand. (d) Fnd he SSE and he average error for he model. (a) P() = (b) P(25) = P(40) = The populaon of San Dego was abou housand n 1985 and wll be abou housand n 2000.

5 5 Fng Exponenal Models o Daa (c) For hs queson, we se P() = 1500 and solve for = = = = = Thus, he populaon of San Dego wll reach 1500 housand n December (d) P P() Error, E 2 E (Acual) (Predced) P P() The SSE = = And he average error s average error =

6 6 Fng Exponenal Models o Daa Exercses: In each of problems 1 and 2 he populaon census daa for a U.S. cy s gven. (a) Fnd an exponenal model for he daa usng he frs and las daa pons. Le = 0 n he year Round he value for P 0 o wo decmal places and he value of a o four decmal places. Use o predc he populaon n Calculae he average error of he model. (b) Fnd he exponenal model ha bes fs hs census daa. Le be 0 n he year Round he value for P 0 o wo decmal places and he value of a o four decmal places. Use o predc he populaon n Calculae he average error of he model. Fnd he annual percenage growh/decay rae. 1. San Anono, Texas: Year Pop. (hous) Buffalo, New York: Year Pop. (hous) As cassee apes and compac dsks became more popular, he sales of vnyl sngles declned, as shown n he followng able. Year Mllons of uns, s (a) (b) Fnd he bes-f exponenal model, s() = s 0 a, for he daa. Le = 0 n Wha s he decay rae n sales of vnyl sngles? Suppose ha vnyl sngles wll be dsconnued when her sales fall below 2 mllon. In wha year wll hs occur? 4. The number of females praccng medcne as MDs s gven n he able below for seleced years Year Number, N, (hous) (a) Fnd he bes-f exponenal model, N() = N 0 a, for he daa. Le = 0 n Wha s he growh rae of female MDs? (b) Wha s he approxmae number of female MDs n 1988? (c) Approxmaely how many female MDs were here n 2005? (d) In wha year wll he number of female MDs exceed 0 housand?

7 7 Fng Exponenal Models o Daa 5. The able below lss hear dsease deah raes per 0,000 people n 2001 for seleced ages. Age, x Deah Rae, R (a) (b) Fnd he bes-f exponenal model, R() = R 0 a x, for he daa. Le x be he acual age. Round boh R 0 and a o four decmal places. Esmae he hear dsease deah rae for people who are 80 years old. Answers: 1. (a) P() = ; P(40) 92 housand; average error (b) P() = ; P(40) 82 housand; average error 13.07; growh rae: 1.59% 2. (a) P() = ; P(40) 278 housand; average error (b) P() = ; P(40) 269 housand; average error 13.15; decay rae: 1.7% 3. (a) s() = ; decay rae: 20.46% per year (b) May (a) N() = ; growh rae: 6.92% per year (b) 85,755 (c) 267,463 (d) Sepember (a) R(x) = x (b) 23 deahs per 0,000