Chapter 1.3, Problem 40E

(a)

To determine

To find: The exponential regression for the given data and also graph it.

Answer to Problem 40E

The exponential regression is $y=23668{\left(1.0236\right)}^x$ . The graph of the exponential regression is shown in Figure (1).

Explanation of Solution

Given information:

The following table gives the population of Texas for several years:

Population of Texas
Year	Population (thousands)
$1980$	$23668$
$1990$	$29811$
$1995$	$31697$
$1998$	$32988$
$1999$	$33499$
$2000$	$33872$

Calculation:

Assume that the exponential regression for population is $y=a{b}^x$ .

As $x=0$ denote the year $1980$ . So, $y$ is equal to $23668$ for $x=0$ . Substitute $0$ for $x$ and $23668$ for $y$ in the exponential regression.

  $\begin{array}{c}y=a{b}^x\\ 23668=a{b}^0\\ a\times 1=23668\\ a=23668\end{array}$

Substitute $14229$ for $a$ in exponential regression.

  $\begin{array}{l}y=a{b}^x\\ y=23668\times b^x\end{array}$

Now, the year $1990$ is denoted by $x=10$ and the population in year $1990$ is $29811$ . Substitute $10$ for $x$ and $29811$ for $y$ in the exponential regression.

  $\begin{array}{c}y=23668\times b^x\\ 29881=23668\times b^{10}\\ b^{10}=\frac{29881}{23668}\\ b={\left(1.2625\right)}^{\frac{1}{10}}\end{array}$

Use the calculator for the value of ${\left(1.2625\right)}^{\frac{1}{10}}$ . Press “ON” button and enter the keystrokes as given below:

  $\boxed{(}\boxed{1.2625}\boxed{)}\boxed{^}\boxed{0.1}$

So, the value of $b$ is equal to $1.0236$ . The exponential regression is $y=23668{\left(1.0236\right)}^x$ .

To graph a function $y=23668{\left(1.0236\right)}^x$ , follow the steps using graphing calculator.

First press “ON” button on graphical calculator, press $\boxed{\text{Y}=}$ key and enter right hand side of the equation $y=23668{\left(1.0236\right)}^x$ after the symbol ${\text{Y}}_1$ . Enter the keystrokes $\boxed{23668}\boxed{(}\boxed{1.0236}\boxed{)}\boxed{^}\boxed{\text{X}}$ .

The display will show the equation,

  ${\text{Y}}_{\text{1}}=23668\left(1.0236\right)^\text{X}$

Now, press the $\boxed{\text{GRAPH}}$ key and $\boxed{\text{TRACE}}$ key to produce the graph of given quadratic equation in standard window as shown below.

  

Figure (1)

Therefore, the exponential regression is $y=23668{\left(1.0236\right)}^x$ . The graph of the exponential regression is shown in Figure (1).

(b)

To determine

To find: The population of Texas in $2003$ with exponential regression and compare it with $35484000$ .

Answer to Problem 40E

The population of Texas in $2003$ with exponential regression is $40472304.75$ . The estimation population exceeds the actual value by $4988304.75$ .

Explanation of Solution

Given information:

The following table gives the population of Texas for several years:

Population of Texas
Year	Population (thousands)
$1980$	$23668$
$1990$	$29811$
$1995$	$31697$
$1998$	$32988$
$1999$	$33499$
$2000$	$33872$

Calculation:

As calculated in part (a), the exponential regression for population is $y=23668{\left(1.0236\right)}^x$ .

The population in $2003$ is represented by $x=23$ . Substitute $x=23$ in the exponential regression for the population in $2003$ .

  $\begin{array}{l}y=23668{\left(1.0236\right)}^x\\ y=23668{\left(1.0236\right)}^{23}\end{array}$

Use the calculator for the value of $23668{\left(1.0236\right)}^{23}$ . Press “ON” button and enter the keystrokes as given below:

  $\boxed{23668}\boxed{(}\boxed{1.0236}\boxed{)}\boxed{^}\boxed{23}$

So, the population of Texas in year $2003$ is $40472304.75$ thousands.

Therefore, the population of Texas in $2003$ with exponential regression is $40472304.75$ . The estimation population exceeds the actual value by $4988304.75$ .

(c)

To determine

To find: The annual rate of growth of population of Texas.

Answer to Problem 40E

The population increase by $2.36\%$ every year.

Explanation of Solution

Given information:

The following table gives the population of Texas for several years:

Population of Texas
Year	Population (thousands)
$1980$	$23668$
$1990$	$29811$
$1995$	$31697$
$1998$	$32988$
$1999$	$33499$
$2000$	$33872$

Calculation:

As calculated in part (a), the exponential regression for population is $y=23668{\left(1.0236\right)}^x$ .

By the exponential regression tells that the population of Texas becomes $1.0236$ times every year.

So, the rate of growth of the population is $2.36\%$ .

Therefore, the population increases by $2.36\%$ every year.