Chapter 3.2, Problem 47E

Solution

The logistic regression model and predict the U.S. pollution in the year 2020.

The predict population in 2020 will be $336.5\text{ million}$ .

Given Information:

The table is defined as,

Year	Population
1900	$76.2$
1910	$92.2$
1920	106
1930	$123.2$
1940	$132.2$
1950	$151.3$
1960	$179.3$

Calculation:

Consider the given polynomial,

By using a graphing calculator, enter the year values starting from $t=0$ for 1900 on ${\text{L}}_1$ and the population in millions on ${\text{L}}_2$ .

Step 1. Press the tables’ option and enter the values as mentioned above.

  

Step 2. Use the Logistic feature to find the logistic regression model.

Display values are following as,

  

Thus, the function can be written as following,

  $\begin{array}{c}y=\frac{702.32}{1+8.0257e^{-0.01666x}}\\ P\left(t\right)=\frac{702.32}{1+8.0257e^{-0.01666t}}\left[\text{Population model}\right]\end{array}$

Substitute 120 for t to find the population in 2020 as $2020-1900=120$ .

  $\begin{array}{c}P\left(120\right)=\frac{702.32}{1+8.0257e^{-0.01666\left(120\right)}}\\ =\frac{702.32}{\text{2}\text{.08703}}\\ \approx 336.5\end{array}$

Therefore, the population in 2020 is $336.5\text{ million}$ and the required model is $P\left(t\right)=\frac{702.32}{1+8.0257e^{-0.01666t}}$ .