Chapter 3.4, Problem 29AYU

Which Model? The following data represent the birth rate (births per 1000 population) for women whose age is $a$, in 2012.

(a) Using a graphing utility, draw a scatter diagram of the data, treating age as the independent variable. What type of relation appears to exist between age and birth rate?

(b) Based on your response to part (a), find either a linear or a quadratic model that describes the relation between age and birthrate,

(c) Use your model to predict the birth rate for 35-year- old women.

To determine

To calculate:

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

b. Find the equation of best fit.

c. Predict the birth rate for 35 year old woman.

Answer to Problem 29AYU

a. The graph is given below.

b. The equation of best fit is $B=-0.5471x^2+31.19x-342.22$.

c. The birth rate for a 35 year old woman is 79 units.

Explanation of Solution

Given:

The given data represents the relation between the birth rate and the age of women.

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 linear relationship with positive slope.

b. The equation of best fit is $B=-0.5471x^2+31.19x-342.22$.

c. When $a=35$ we get

$B=-0.5471{(35)}^2+31.19(35)-342.22=79.2325$.

The birth rate for a 35 year old woman is 79 units.