nathalie-alex-sojung-vincci

=**Population in the US per Square Mile**=


 * Year || People per Square Mile ||
 * 1790 || 4.5 ||
 * 1800 || 6.1 ||
 * 1810 || 4.3 ||
 * 1820 || 5.5 ||
 * 1830 || 7.4 ||
 * 1840 || 9.8 ||
 * 1850 || 7.9 ||
 * 1860 || 10.6 ||
 * 1870 || 10.9 ||
 * 1880 || 14.2 ||
 * 1890 || 17.8 ||
 * 1900 || 21.5 ||
 * 1910 || 26 ||
 * 1920 || 29.9 ||
 * 1930 || 34.7 ||
 * 1940 || 37.2 ||
 * 1950 || 40.6 ||
 * 1960 || 50.6 ||
 * 1970 || 57.5 ||
 * 1980 || 64 ||

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) =ab^tFor points (0, 4.5)ab^0 a= 4.5 f(t) = 4.5b^t First turn the year into the difference between x2 and x1 so the middle point is (100, 17.8) 17.8 = 4.5 b^100 17.8/4.5 =?? b log 3.956/100= log b log 3.956 =?? 100 log b 0.00597= log b b = 10^0.00597 y = 4.5 x 1.0138^t b approx. 1.0138

2) ExpReg y = a * b^nx a = 3.9917 b = 1.015 r^2 = 0.98 r = 0.99

{INSERT RESIDUAL PLOT HERE & window}

3) Both functions are a good model, however the one that fits the data more accurately is y = 3.9917 * 1.015^t. This is because the one that was done by hand only used 2 points, while the equation shown by the calculation used all the points - therefore it is more precise.

4) 4.5 * 2 = 4.5 * 1.0138^t This means it would take 50.57 years to double the people per square mile. log 2= t log 1.0138 log 2 / log 1.0138 = t t = 50.57 years

5)

6) y = 3.9917 * 1.015^t is a better equation, because it used all the points to make the equatoin, instead of 2 po ints, therefore it is more precise. =Part II=

LINEARIZING DATA (refer to math textbook pg 363) y = 3.9917 x 1.015^t log y = t log 1.015 + log 3.9917 log y = 0.601 + 0.00646t log y =y t= x y = 0.00646x + 0.601 (x, log y) is the linearizing factor

=Part III= 1) Graph a) (x,y)

Graph 2) (log x, y)

Graph 3) (x, log y)

Graph 4) (log x, log y)

Graphs 3 and 4 linearize our data the best.

2) As you can see, Graphs 1 and 2 look quite similar. However r^2 of Graph 1 (x,y) is 0.89, and the r^2 of Graph 2 (log x,y) is 0.88. Graphs 3 and 4 are the same. They both have an r^2 value of 0.98. (All r^2 values can be found by making a least squares line) = =