World+Population+over+the+past+2.5+millennium.

Group: Michael Yao, Jessie ho, Nathaniel, and Avani

Data For world population growth:
 * Year (BC) || Population in Mill ||
 * -400 || 162 ||
 * -200 || 231 ||
 * 0 || 255 ||
 * 200 || 256 ||
 * 400 || 206 ||
 * 500 || 206 ||
 * 600 || 206 ||
 * 700 || 207 ||
 * 800 || 224 ||
 * 900 || 226 ||
 * 1000 || 254 ||
 * 1100 || 301 ||
 * 1200 || 400 ||
 * 1250 || 416 ||
 * 1300 || 432 ||
 * 1340 || 443 ||
 * 1400 || 374 ||
 * 1500 || 460 ||
 * 1600 || 579 ||
 * 1700 || 679 ||
 * 1750 || 770 ||
 * 1800 || 954 ||
 * 1850 || 1,241 ||
 * 1900 || 1,633 ||
 * 1950 || 2,527 ||

I choose the exponential because it logically makes sense that human population overall would be growthing at a constant rate. When graphed, the data looks quite clean without any major outliers.

A - Perhaps You Have an Exponential Model
Find an exponential model ( f (t) = ab^t ) to fit your data set in two ways:

1) f(t) (by hand)

Choose two points for my T and T+1 I choose the points (1400,374.0) and (1500,460.0) We now assume that starting from 1400 the population grew at a constant rate per year, and now we're trying to figure out what that constant is.

we know that 460.0/374.0 is the growth rate over 100 years. So 460/374=T^100. Now it's really easy.

460/374=T^100

(460/374)^-100=T T= 1.0020718502424824353407259876337

Double Check 1.0020718502424824353407259876337^100=460/372

The constant growth rate is 1.00207185

Now that we know the constant growth rate, we need to find the constant multiplier. Knowing that when T=0, y=255 so 255 is our constant multiplier.

However, i then noticed that y=255 for the next thousand years until the year 1150. I then assumed that this ment the population didn't start rising since 1000, so i changed my equation to adjust

The final equation is

y=255(1.0020718502424824353407259876337)^t

2. I put the data into the calc and ran expodential reg. The equation turns to be 158.1197832*1.000895996^X

3. I think that both of the models were valid for specific areas of time. The cacl model was more accurte predicting the first 1700 years, however, the hand done method fits the last 200 years growth.

4. one of the predictions from the model we can look at is the time human population was yero. Typing in the second cacl found equation, i also put in y=o and seached for intercept.

5.

PART II For this set of data, we use two method to linearize the graph, one is med med line and another one is least square line. Least square line: Med med line:

Compare two graphs, we think that the least square line is better. As the least square line shows the average growth rate of the data points. It gives the trend of the data.

PART III

At first, we tried all different combinations of x, y, log x, log y, ln x, ln y.

Although we tried all the different combinations, we could not find a solution that linearized our data.