   Chapter 14, Problem 35RE 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 US. average wage The table gives the average annual U.S. wage from 2011 and projected through 2021.(a) Use linear regression to find the best linear model for the wage as a function of the number of years after 2010. Report the model with 4 significant digits.(b) Use the reported model to predict the average annual wage in 2025. Year Wage ($) Year Wage ($) 2011 43,009 2017 55,989 2012 44,644 2018 58,698 2013 46,496 2019 61,179 2014 48,595 2020 63,676 2015 50,893 2021 66,161 2016 53,317 Source: Social Security Administration

(a)

To determine

To calculate: The linear regression to find the best linear model for the wage as a function of the number of years after 2010 where the table gives the average annual U.S. wage from 2011 and projected through 2021.

 Year Wage ($) 2011 43,009 2012 44,644 2013 46,496 2014 48,595 2015 50,893 2016 53,317 2017 55,989 2018 58,698 2019 61,179 2020 63,676 2021 66,161 Explanation Given Information: The table gives the average annual U.S. wage from 2011 and projected through 2021.  Year Wage ($) 2011 43,009 2012 44,644 2013 46,496 2014 48,595 2015 50,893 2016 53,317 2017 55,989 2018 58,698 2019 61,179 2020 63,676 2021 66,161

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 denote the years past 2010 for 1<i<11.

That is, x1=1,x2=2,x3=3,x4=4, x5=5,x6=6,x7=7,x8=8, x9=9,x10=10,x11=11.

Let yi denote the wages for 1<i<11.

That is,

y1=43,009,y2=44,644,y3=46,496,y4=48,595, y5=50,893,y6=53,317,y7=55,989,y8=58,698, y9=61,179,y10=63,676,y11=66,161.

Since, the number of data points is 11.

Thus, n=11.

Calculate the value of b.

Substitute 11 for n in b=i=1nxii=1nyini=1nxiyi(i=1nxi)2ni=1nxi2.

b=i=111xii=111yi11i=111xiyi(i=111xi)211i=111xi2

That is,

b=(x1+x2+x3+x4+x5+x6+x7+x8+x9+x10+x11)(y1+y2+y3+y4+y5+y6+y7+y8+y9+y10+y11)11(x1y1+x2y2+x3y3+x4y4+x5y5+x6y6+x7y7+x8y8+x9y9+x10y

(b)

To determine

To calculate: The average annual wage in 2025 using the reported model where the table gives the average annual U.S. wage from 2011 and projected through 2021.

 Year Wage (\$) 2011 43,009 2012 44,644 2013 46,496 2014 48,595 2015 50,893 2016 53,317 2017 55,989 2018 58,698 2019 61,179 2020 63,676 2021 66,161

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