   Chapter 14, Problem 36RE 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

Modeling Civilian workforce The following table gives the number of women and the number of men (both in millions) in the U.S. civilian labor force for selected years from 1950 and projected to 2050.(a) Let x represent the millions of women and y represent the millions of men and find the linear regression model for y = f ( x ) . Report the model to 3 significant digits.(b) Find and interpret the slope of the reported model.U.S. Civilian Workforce (in millions): 1950-2050 Women Men Women Men 18.4 43.8 45.5 61.5 23.2 46.4 56.8 69.0 31.5 51.2 65.7 75.2 Women Men Women Men 75.5 82.2 81.6 88.5 78.6 84.2 86.5 94.0 79.3 85.4 91.5 100.3 Source: U.S. Bureau of Libor Statistics

(a)

To determine

To calculate: The linear regression model for y=f(x) where x represents the millions of women and y represents the millions of men for the following table that gives the number of women and the number of men (both in millions) in the U.S. civilian labor force for selected years from 1950 and projected to 2050.

 Women Men 18.4 43.8 23.2 46.4 31.5 51.2 45.5 61.5 56.8 69.0 65.7 75.2 75.5 82.2 78.6 84.2 79.3 85.4 81.6 88.5 86.5 94.0 91.5 100.3
Explanation

Given Information:

The provided table gives the number of women and the number of men (both in millions) in the U.S. civilian labor force for selected years from 1950 and projected to 2050.

 Women Men 18.4 43.8 23.2 46.4 31.5 51.2 45.5 61.5 56.8 69.0 65.7 75.2 75.5 82.2 78.6 84.2 79.3 85.4 81.6 88.5 86.5 94.0 91.5 100.3

Formula used:

According to linear regression equation, the equation of the line that is the best fit 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:

Let xi represents the millions of women for 1<i<12.

That is, x1=18.4,x2=23.2,x3=31.5,x4=45.5, x5=56.8,x6=65.7,x7=75.5,x8=78.6, x9=79.3,x10=81.6,x11=86.5,x12=91.5.

Let yi represents the millions of men for 1<i<12.

That is, y1=43.8,y2=46.4,y3=51.2,y4=61.5, y5=69.0,y6=75.2,y7=82.2,y8=84.2, y9=85.4,y10=88.5,y11=94.0,y12=100.3.

Since, the number of data points is 12.

Thus, n=12.

Calculate the value of b.

Substitute 12 for n in b=i=1nxii=1nyini=1nxiyi(i=1nxi)2ni=1nxi2.

b=i=112xii=112yi12i=112xiyi(i=112xi)212i=112xi2

That is,

b=(x1+x2+x3+x4+x5+x6+x7+x8+x9+x10+x11+x12)(y1+y2+y3+y4+y5+y6+y7+y8+y9+y10+y11+y12)12(x1y1+x2y2+x3y3+x4y4+x

(b)

To determine

To calculate: The slope of the reported model y=f(x) where x represents the millions of women and y represents the millions of men for the following table that gives the number of women and the number of men (both in millions) in the U.S. civilian labor force for selected years from 1950 and projected to 2050 and to interpret the meaning. The following table gives the number of women and the number of men (both in millions) in the U.S. civilian labor force for selected years from 1950 and projected to 2050.

 Women Men 18.4 43.8 23.2 46.4 31.5 51.2 45.5 61.5 56.8 69.0 65.7 75.2 75.5 82.2 78.6 84.2 79.3 85.4 81.6 88.5 86.5 94.0 91.5 100.3

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