Chapter 3.2, Problem 50E

Solution

The logistic regression model and check that it is similar to the exercise 56 of 3.1 section.

Both are similar.

Given Information:

The table is defined as,

Year	Arizona
1900	$0.1$
1910	$0.2$
1920	$0.3$
1930	$0.4$
1940	$0.5$
1950	$0.7$
1960	$1.3$
1970	$1.8$
1980	$2.7$
1990	$3.7$
2000	$5.1$

Calculation:

Consider the given polynomial,

By using a graphing calculator, enter the year values from $t=100$ for 1900 (as t represents the number of years since 1800) on ${\text{L}}_1$ and the population of New York (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,

  $\begin{array}{c}y\approx \frac{15.642}{1+11799.36e^{-0.04324x}}\\ P\left(t\right)=\frac{15.642}{1+11799.36e^{-0.04324t}}\left[\text{Population model}\right]\end{array}$

Now, write the mode as obtained in the mentioned exercise 56 of section 3.1 of this book.

  $P\left(t\right)=\frac{15.642}{1+11799.36e^{-0.04324t}}$

Thus, both the modes are similar.

The limit to growth of New York is about 19.875 million while the limit to growth of Arizona is about $15.642$ which means that the population of Arizona cannot surpass that of New York.

Further prove this by graphing both models in the same window where they never intersect:

Now, set the window of the calculator as following,

  

Now, use plot key and the displaying graph will be looks like this,

  

Thus, the model matches.

Therefore, the population of Arizona cannot surpass to New York population.