Growth+of+Bluegills


 * Initial Length (mm) || Total Length after 1 Year (mm) ||
 * 48 || 69 ||
 * 52 || 71 ||
 * 51 || 69 ||
 * 53 || 75 ||
 * 69 || 101 ||
 * 71 || 107 ||
 * 69 || 100 ||
 * 75 || 104 ||
 * 101 || 138 ||
 * 107 || 138 ||
 * 100 || 130 ||
 * 104 || 140 ||
 * 138 || 160 ||
 * 138 || 157 ||
 * 130 || 156 ||
 * 140 || 161 ||
 * 160 || 173 ||
 * 157 || 168 ||
 * 156 || 172 ||
 * 161 || 178 ||
 * 173 || 176 ||
 * 168 || 174 ||
 * 172 || 173 ||
 * 178 || 178 ||

As you can see this is an inverse(Logarithmic) Function Part I Question 1 get the equation for the function Choose two points to represent the data (75,53) & (101,69) ab^x=y ab^75=53, ab^101=69 ab^101=69 Seperate b into b^75*b^26 instead of b^101 ab^75*b^26=69 Sub ab^75 to 53 53*b^26=69 b=(69/53)^(1/26)=1.010198368 Sub b into ab^x=y and use 75,53 a*1.010198368^75=53 a=53/2.140423332=24.76145686 24.76145686*1.010198368^x=y Flip x and y so you can undo the inverse 24.76145686*1.010198368^y=x y=24(1.01)^x x=24(1.01)^y log x/24 = log 1.01^y log x - log 24 = y log 1.01 1/log 1.010198368 * log x - log 24.76145686 /log 1.010198368 = y 1/log b * log x - log a /log b = y 1/log b = B log a /log b = A B log x - A = y This is the form of for the inverse(log) equation

ExpReg = 22.067169052*1.0115673601^x Change this into ab^x and sub it into the equation 1/log b * log x - log a /log b = y 1/log 1.0115673601 * log x - log 22.067169052 /log 1.0115673601 = y

Question 2 Make a Graph Inverse Function of Bluegills

Question 3 Find the doubling time 69th year = 49.8696029 Double time = 69 years
 * x || y ||
 * 0 || 24.76145686 ||
 * 1 || 25.01398 ||
 * 2 || 25.269085 ||
 * 69 || 49.8696029 ||
 * 69 || 49.8696029 ||

Question 4 Use LnReg y=.266.28899973318+86.361288896291 ln(x) y=a+bln(x) lnex = log x/log e y=a+b (log x/log e) y= -266.28899973318+86.361288896291 (log x/log e) y= -266.28899973318+86.361288896291/log e * log x This is the form of a+b log x therefore it also fits ExpReg

Part II In this part, x is initial length while y is total length after 1 yr Graph: Log x vs log y: Log x vs y: Log y vs x:

From the above graphs, the best linear graph is the log x vs y. As when we plot the residual plot, (see below), all the residual points are distributed with no patterns and also, compare with the other two graphs, the trend of graph log x vs y is exactly a linear equation. Residual plot:

Calculation of equation: y = a log x + b y = 198.8542164 log x + -266.2889997 log 10^y = log x^(198.8542164) + log 10^(-266.2889997) log 10^y = log (x^198.8542164 * 10^-266.2889997) 10^y =10^-266.2889997 * x^198.8542164 ,,

Part III



The graph that best linearizes our data is the graph of Log X, Y. If this was graphed on paper, it would be graphed using Semi-Log paper. The graph only touches a few points, but it is the best fit line, with a coefficient correlation of 99%.