Chapter 3.4, Problem 25AYU

Life Cycle Hypothesis An individual’s income varies with his or her age. The following table shows the median income I of males of different age groups within the United States for 2012. For each age group, let the class midpoint represent the independent variable, $x$. For the class “65 years and older,” we will assume that the class midpoint is $69.5$.

(a) Use a graphing utility to draw a scatter diagram of the data. Comment on the type of relation that may exist between the two variables.

(b) Use a graphing utility to find the quadratic function of best fit that models the relation between age and median income.

(c) Use the function found in part (b) to determine the age at which an individual can expect to earn the most income.

(d) Use the function found in part (b) to predict the peak income earned.

(e) With a graphing utility, graph the quadratic function of best fit on the scatter diagram.

To determine

To calculate:

a. Graph a scattered diagram and determine the type of relation that exists between the 2 variables.

b. Find the quadratic function of best fit.

c. Determine the age at which the individual gets maximum income.

d. Find the maximum income earned.

e. Graph the quadratic function of best fit.

Answer to Problem 25AYU

a. The graph is given below.

b. The equation of best fit is $\$47649.464$

c. People can expect a maximum income at around the age of 48.

d. The maximum income earned is about $\$47649.464$.

e. The graph is given below.

Explanation of Solution

Given:

The median income, $I$ of males of different age groups is given below. The class midpoint of the age group represents the $x$ variables.

Formula used:

For a quadratic equation $y=a{x}^2+bx+c$, if $a$ is positive, then the vertex $\left(\frac{-b}{2a},<mtext> </mtext>y\left(\frac{-b}{2a}\right)\right)$ is the minimum point and if $a$ is negative, the vertex $\left(\frac{-b}{2a},<mtext> </mtext>y\left(\frac{-b}{2a}\right)\right)$ is the maximum point.

Calculation:

We can draw the scatter diagram using Microsoft Excel.

Thus, on entering the $x$ and the $y$ values on excel, we have to choose the Scatter diagram form the insert option.

Then for getting the equation of best fit, we have to choose the Layout option and then Trendline and then more Trendline option. Then choose the option polynomial and then display the equation.

Thus, we get the scatter diagram with the equation of best fit as

a. The scatter diagram is drawn above and we can see that the given variables exhibit a non-linear polynomial relationship.

b. The equation of best fit is $y=-44.759x^2+4295.4x-55045$

c. We can see that the equation of best fit is a quadratic equation with $a=-44.759,<mtext> </mtext>b=4295.4,<mtext> </mtext>c=55045$

Here,$a$ is negative, therefore, the vertex is the maximum point.

The $x$ value for maximum $y$ is

$\begin{array}{l}x=-\frac{b}{2a}=-\frac{4295.4}{2(-44.759)}\\ = 47.98\approx 48\end{array}$

Thus, at around the age of 48, people can expect a maximum income.

d. The maximum income earned is

$y(48)=-44.759{(48)}^2+4295.4(48)-55045=47649.464$

The maximum income earned is about $\$47649.464$.

e. The graph is drawn above.