Question

Men and women have lived in poverty since the beginning of time. Even today millions of people live below the poverty line. Source:US Census Bureau."Table 7. Poverty in People, by Sex." 1. Use the table below to answer the questions. Year 2000 (t=0) 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 Men Below Poverty Line (in millions) 136.274 137.558 139.558 140.931 142.433 143.803 145.486 146.655 147.562 149.237 149.737 Women Below Poverty Line (in millions) 142.670 143.917 145:759 146.768 148.183 149.331 150.964 152.044 153.179 154.582. 156.394 a. Use linear regression to model the number of men, M, as a function of time, t. b. Use linear regression to model the number of women. W. as a function of time, t. C. What are the rates of change for each function? What do they mean? d. What are the vertical intercepts for each function? What do they mean?!

Answer 1

Solution:

Given: Tabular Form for number of Men and Women under poverty line in different Year>

Year	Men	Women
2000	136.274	142.67
2001	137.558	143.917
2002	139.558	145.759
2003	140.931	146.768
2004	142.433	148.183
2005	143.803	149.331
2006	145.486	150.964
2007	146.655	152.004
2008	147.562	153.179
2009	149.237	154.582
2010	149.737	156.394

We have to find Line Regression Model for Men and Women as a function of time

If M,W represents dependent variable ie No. of Men, Women under poverty line

and t represents the year

we have

Regression Model as

M=a+bt

where

a = [(∑y)(∑x²)-(∑x)(∑xy)] / [n(∑x²)-(∑x)²]

b = [n(∑xy)-(∑x)(∑y)] / [n(∑x²)-(∑x)²]

solution for part (a) Linear regression model for men M as a function of time t

year(t)	men(M)	t*M	t^2	M^2
2000	136.274	272548	4000000	18570.6
2001	137.558	275253.6	4004001	18922.2
2002	139.558	279395.1	4008004	19476.44
2003	140.931	282284.8	4012009	19861.55
2004	142.433	285435.7	4016016	20287.16
2005	143.803	288325	4020025	20679.3
2006	145.486	291844.9	4024036	21166.18
2007	146.655	294336.6	4028049	21507.69
2008	147.562	296304.5	4032064	21774.54
2009	149.237	299817.1	4036081	22271.68
2010	149.737	300971.4	4040100	22421.17
Sum	1579.234	3166517	44220385	226938.5

so with formula for a and b ie

a = [(∑y)(∑x²)-(∑x)(∑xy)] / [n(∑x²)-(∑x)²]

b = [n(∑xy)-(∑x)(∑y)] / [n(∑x²)-(∑x)²]

we have a= -2636.89 and b= 1.39

the regression model is

M=-2636.89 + 1.39t

solution for part (b) Linear regression model for Women M as a function of time t

year(t)	women(W)	W*t	t^2	W^2
2000	142.67	285340	4000000	20354.73
2001	143.917	287977.9	4004001	20712.1
2002	145.759	291809.5	4008004	21245.69
2003	146.768	293976.3	4012009	21540.85
2004	148.183	296958.7	4016016	21958.2
2005	149.331	299408.7	4020025	22299.75
2006	150.964	302833.8	4024036	22790.13
2007	152.004	305072	4028049	23105.22
2008	153.179	307583.4	4032064	23463.81
2009	154.582	310555.2	4036081	23895.59
2010	156.394	314351.9	4040100	24459.08
Sum	1643.751	3295868	44220385	245825.1

so with formula for a and b ie

a = [(∑y)(∑x²)-(∑x)(∑xy)] / [n(∑x²)-(∑x)²]

b = [n(∑xy)-(∑x)(∑y)] / [n(∑x²)-(∑x)²]

we have a= -2526.20 and b= 1.33

the regression model is

M=-2526.20 + 1.33t

solution for part (c)

Rate of change for function from regression model comparing with y = mx + c, [ where m is slope(rate of change and c is intercept] is b

Rate of change for function for Men is 1.39

Rate of change for function for Women is 1.33

This shows that every year 1.39 men and 1.33 women goes below poverty line.

solution for part (d)

Intercept for function from regression model(comparing with y = mx + c, [where m is slope(rate of change) and c is intercept]is a

Intercept for function for Men is -2636.89

Rate of change for function for Women is -2526.20

This shows that at starting ie t=0 there were no men and women below poverty line(as number of men and women cannot be negative).