Let p(y)=aeby+c. When y=1970 we want p(1970)=219.5, so if a=219.5 then 1970b+c=0, c=-1970b.
So p(y)=219.5eby-1970b or 219.5eb(y-1970).
Now we have to find b. Let’s take the population of two consecutive decades:
y=1970, p(1970)=219.5.
y=1980, p(1980)=268.0.
p(1980)/p(1970)=e1980b+c/e1970b+c=e10b=268/219.5=1.2210.
10b=ln(1.221)=0.2, so b=0.02.
We have taken the ratio of the populations of 1970 and 1980. We need to consider populations from other years to be sure there’s constant growth.
So 326.5/268=1.2183, 397.5/326.5=1.2175, 485/397.5=1.2201.
They’re all fairly close to one another, so b=0.02 is a fair estimate.
We could simply take logs of each population and compare them:
ln(219.5)=5.3914, ln(268)=5.5910, ln(326.5)=5.7884, ln(397.5)=5.9852, ln(485)=6.1841.
There’s a common difference of about 0.20, but if we take the average difference we get 0.1982. ln(1.2192)=0.1982=10b, b=0.01982, so the average ratio of the decade populations is 1.2192. Let’s see what b=0.02 predicts (to the nearest 0.5):
1970 219.5
1980 268.0
1990 327.5
2000 400.0
2010 488.5
2015 540.0
Compare with b=0.01982:
1970 219.5
1980 267.5
1990 326.5
2000 398.0
2010 485.0
2015 535.5
The second model fits better than the first, so:
(a) p(y)=219.5e0.01982(y-1970) is a good model.
(b) prediction for 2015 using model in (a) is population (thousands) = 535.5
Another way of looking at this problem is to consider a geometric series where the first term is a, the next term is ar, then ar2, ar3, etc. So a would be 219.5 and r would be the average increasing rate. The data shows decades and we are asked to predict the population for 2015, which is not a decade. So r has to be the rate of increase of no more than 5 years. If we make r the annual increase, then the series would be a, ar10, ar20, ar30, etc. But we need to relate these exponents to the actual year, so we get ary-1970. We already know that r10=1.2192. From this 10ln(r)=ln(1.2192)=0.1982, so ln(r)=0.01982, making r=1.02 (about 2% per year). So p(y)=219.5(1.02)y-1970. Lets try it out:
1970 219.5
1980 267.5
1990 326.0
2000 397.5
2010 484.5
2015 535.0
(To the nearest 0.5 thousands, that is, to the nearest 500). The model fits well.