Chapter 7, Problem 65RE

a.

To determine

To Find:

To find a curve to model the data and superimpose it on a scatter plot of population against years after 1950.

Answer to Problem 65RE

The graph is shown superimposed

  

Explanation of Solution

Given information:

The given data is below:

  

Formula:

The general logistic regression equation is

  $P=\frac{M}{1+A{e}^{-\left(Mk\right)t}}$ where $A$ -constant determined by an appropriate initial condition;

  $M$ -carrying capacity; $k$ -growth constant.

The regression equation is given by $P=\frac{M}{1+A{e}^{-\left(Mk\right)t}}$

Now substituting the values,

  $P=\frac{272286.352}{1+302.687e^{-\left(0.209\right)t}}$

Graph:

  

Interpretation:

The graph is shown superimposed on the scatter plot. The fit is almost perfect.

Therefore, the graph is shown superimposed.

b.

To determine

To Find:

To find the number of Anchorage population approach in the long run.

Answer to Problem 65RE

The number of Anchorage population approach in the long run is $272,286$ people.

Explanation of Solution

Given information:

The given data is below:

  

The regression equation is given by $P=\frac{M}{1+A{e}^{-\left(Mk\right)t}}$

Now substituting the values,

  $P=\frac{272286.352}{1+302.687e^{-\left(0.209\right)t}}$

Therefore, the number of Anchorage population approach in the long run is $272,286$ people.

c.

To determine

To Find:

To find the logistic differential equation in the form $\frac{dP}{dt}=kP\left(M-P\right)$ .

Answer to Problem 65RE

The logistic differential equation is $\frac{dP}{dt}=\left(7.67\times {10}^{-7}\right)P\left(272286-P\right)$ .

Explanation of Solution

Given information:

The given data is below:

  

The logistic differential equation is $\frac{dP}{dt}=kP\left(M-P\right)$

So from part (a) the carrying capacity $M=272286$ ,

  $\begin{array}{l}Mk=0.209\\ k=\frac{0.209}{272286}\\ k\approx 7.67\times {10}^{-7}\end{array}$

Now substituting the values in the differential equation,

  $\frac{dP}{dt}=\left(7.67\times {10}^{-7}\right)P\left(272286-P\right)$

Therefore, the logistic differential equation is $\frac{dP}{dt}=\left(7.67\times {10}^{-7}\right)P\left(272286-P\right)$ .

d.

To determine

To Find:

To find whether the data is affected by carrying capacity predicted by the regression equation.

Answer to Problem 65RE

There is no reason to be concerned, unless the model application was extremely sensitive.

Explanation of Solution

Given information:

The given data is below:

  

With the new point, the carrying capacity of Anchorage is predicted to be $272,286$ people. This is more than $2\%$ different from the carrying capacity predicted in part (c).

And thus, there is no reason to be concerned, unless the model application was extremely sensitive.