Find the equation of F from table

Question

1988	350
1992	138
1996	33
2000	4
2005	3.2
2007	1.3

Let f(t) be the number of polio cases, in thousands, in the world t years since 1980.Find the equation of F

F(T)=

Answer 1

The easiest way to find the equation for F is to use a calculator or Desmos to do a regression.

You will recognize that the best regression is one using an exponential.

Also, recognizing that year 1988 is the starting year you can set this year equal to 0 to simplify factors that must be determined. (Remember you'll have to revise this at the end to get a correlation to the actual year.)

Place the table in Desmos and replace 1988 with 0 (up to to 19 for 2007), The columns will be automatically labeled x1 and y1. Then enter an exponential correlation to best fit the data as follows;

y₁~ab^x1

This will produce the best regression equation with the constant values.

a = 352.572; b = 0.771875

So the best fit is:

Since x in this equation started at 0 in lieu of 1988, you can use x to represent the actual year if you subtract 1988.

Answer 2

If you plot the points, the curve is an exponential decline, so need to fit the points to

F(x)=y=ab^x using least squares.

First, rewrite the data by taking the natural log of the y points

x ln(y)

8 5.858

12 4.927

16 3.497

20 1.386

25 1.163

27 0.262

Then find m and y₀ using the least square equations for these data points.

Finally, b = e^m and a=e^y0

F(T) = ab^T

I got F(T) = 2798.15(0.771875)^T