   Chapter 14.4, Problem 35E Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781305108042

Solutions

Chapter
Section Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781305108042
Textbook Problem

World population The table gives the actual or projected world population in billions for selected years from 2000 to 2050.(a) Use linear regression to find the linear equation that is the best fit for the data, with x equal to the number of years past 2000.(b) Use the equation to predict the population in 2018.(c) Write a sentence that interprets the slope of the linear regression line. Years Population (billions) 2000 6.08 2010 6.82 2020 7.54 2030 8.18 2040 8.72 2050 9.19 Source: U.S. Bureau of the Census, International Data Base

(a)

To determine

To calculate: The linear equation of the line that is the best fit for the data using linear regression, with x equal to the number of years past 2000. Report the model with 3 significant digits. The world population in billion is tabulated below for their respective years.

 Years Population (in billions) 2000 6.08 2010 6.82 2020 7.54 2030 8.18 2040 8.72 2050 9.19
Explanation

Given Information:

The world population in billion is tabulated below for their respective years.

 Years Population (in billions) 2000 6.08 2010 6.82 2020 7.54 2030 8.18 2040 8.72 2050 9.19

Formula used:

The equation of the best fit line by linear regression that for the given data points (x1,y1),(x2,y2),......,(xn,yn) is y^=a+bx,

Where b=i=1nxii=1nyini=1nxiyi(i=1nxi)2ni=1nxi2, and a=i=1nyibi=1nxin.

Calculation:

Consider the table,

 Years Population (in billions) 2000 6.08 2010 6.82 2020 7.54 2030 8.18 2040 8.72 2050 9.19

Let xi denote the years past 2000 for 1<i<6.

That is, x1=0,x2=10,x3=20,x4=30, x5=40,x6=50.

Let yi denote the population for 1<i<6.

That is, y1=6.08,y2=6.82,y3=7.54,y4=8.18, y5=8.72,y6=9.19.

Since, the number of data points is 6. Thus, n=6.

The value of b for a best fit line from a linear regression is,

Substitute 6 for n in b=i=1nxii=1nyini=1nxiyi(i=1nxi)2ni=1nxi2,

(b)

To determine

To calculate: The population in 2025. The world population in billion is tabulated below for their respective years.

 Years Population (in billions) 2000 6.08 2010 6.82 2020 7.54 2030 8.18 2040 8.72 2050 9.19

(c)

To determine

The interpretation of the slope of the best fit line. The table gives the actual or projected world population in billions for selected years from 2000 to 2050.

 Years Population (in billions) 2000 6.08 2010 6.82 2020 7.54 2030 8.18 2040 8.72 2050 9.19

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