Fit a power function to the data and use it to estimate the number of childbirths (rounded to the nearest whole number) in 2013.

Question

The number of yearly childbirths for a growing town are recorded in the table below. Suppose the number of childbirths each year is a function of the number of years since 1996. Fit a power function to the data and use it to estimate the number of childbirths (rounded to the nearest whole number) in 2013.

Year	1999	2001	2005	2006	2009
Childbirths	13	25	47	51	72

Answer 1

If x=years since 1996 and y=# of childbirths, then our data points are (3,13), (5,25), (9,47), (10,51), and (13,72), and we wish to fit the power function y=ax^b to this data.

This question is easier to solve by transforming it using logarithms, and first solving the problem of fitting the function ln(y) = ln(ax^b) to the data (ln(3),ln(13)), (ln(5),ln(25)), (ln(9),ln(47)), (ln(10),ln(51)), and (ln(13),ln(72)).

Using properties of logarithms, we rewrite the function:

ln(y) = ln(ax^b)

ln(y) = ln(a) + ln(x^b)

ln(y) = ln(a) + b·ln(x)

If we let Y=ln(y), A=ln(a), and X=ln(x), then we have the linear function Y=A+bX.

Now we can do linear regression and fit the line of least squares to the data. You can use a calculator (using STAT>CALC>LinReg(a+bx) on a TI-84), Excel (using =LINEST()), or similar software, or do the calculations manually by using the formulas

slope: b=(n∑x_iy_i – ∑x_i∑y_i)/(n∑x_i² – (∑x_i)²)

Y-intercept: A = (∑y_i – b∑x_i)/n

where n is the number of data points (so n=5 in this question).

We find b=1.1441 and A=1.3323. Now we have the line Y=1.3323+1.1441X fit to the data

(ln(3),ln(13)), (ln(5),ln(25)), (ln(9),ln(47)), (ln(10),ln(51)), and (ln(13),ln(72)).

To return to the original question, we undo the logarithm by applying its inverse:

e^Y = e^{1.3323+1.1441X}

e^{ln y} = e^1.3323e^{1.1441ln x}

y = e^1.3323e^{ln x^1.1441}

y = 3.7897x^1.1441

Now this power function fits the original data (3,13), (5,25), (9,47), (10,51), and (13,72).

2013 is 17 years after 1996. With x=17, we get a prediction of y = 3.7897·17^1.1441 ≈ 97 childbirths.