Chapter 4, Problem 16PT

a.

To determine

Write the slope-intercept form of the equation for the line of fit.

Answer to Problem 16PT

The equation for the line of fit $y=216.69827x-377.465517$ .

Explanation of Solution

Given information: The data where x represents the number of years since 2000 and y

represents the number of Children.

Years x	5	7	9	10	11
Number of children y	442	1254	2275	2511	1732

Calculation:

Formula used:

The equation for the regression line is given below.

  $\begin{array}{c}y=ax+b.\\ \text{Where},a=\frac{n{\displaystyle \sum xy-({\displaystyle \sum x)({\displaystyle \sum y)}}}}{n{\displaystyle \sum x^2-({\displaystyle \sum x{)}^2}}}.\\ b=\frac{{\displaystyle \sum y-a{\displaystyle \sum x}}}{n}.\\ \text{Where, }\text{n=number}\text{of}\text{observations}.\end{array}$

Let x be the number of years since 2000.

Years x	5	7	9	10	11
Number of children y	442	1254	2275	2511	1732

Use a graphing calculator for above table ,

Line of Best Fit

  $\begin{array}{l}y=216.69827x-377.465517.\\ \text{Intercept}=-377.465517.\\ \text{Slope}=216.69827.\end{array}$

b.

To determine

Predict the number of children from E who will be adopted in 2025.

Answer to Problem 16PT

Number of children in 2025 will be $y\approx 5040.$

Explanation of Solution

Given information: The data where x represents the number of years since 2000 and y

represents the number of Children.

Years x	5	7	9	10	11
Number of children y	442	1254	2275	2511	1732

Calculation:

Formula used:

The equation for the regression line is given below.

  $\begin{array}{c}y=ax+b.\\ \text{Where},a=\frac{n{\displaystyle \sum xy-({\displaystyle \sum x)({\displaystyle \sum y)}}}}{n{\displaystyle \sum x^2-({\displaystyle \sum x{)}^2}}}.\\ b=\frac{{\displaystyle \sum y-a{\displaystyle \sum x}}}{n}.\\ \text{Where, }\text{n=number}\text{of}\text{observations}.\end{array}$

Let x be the number of years since 2000.

Years x	5	7	9	10	11
Number of children y	442	1254	2275	2511	1732

Use a graphing calculator for above table ,

Line of Best Fit

  $\begin{array}{l}y=216.69827x-377.465517.\\ \text{Intercept}=-377.465517.\\ \text{Slope}=216.69827.\end{array}$

For predict the number of children from E who will be adopted in 2025 putting $x=2025-2000=25$ in equation of line of best fit (1).

  $\begin{array}{l}y=216.69827x-377.465517.\\ x=25\\ y=216.69827\times 25-\mathrm{377.465517.}\\ y=5417.45675-\mathrm{377.465517.}\\ y=\mathrm{5039.991233.}\\ y\approx 5040.\end{array}$