5.3: Corequisite- Trendlines
- Page ID
- 148618
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)SPECIFIC OBJECTIVES
By the end of this lesson, you should understand that
- linear equations can approximate nearly linear data.
By the end of this lesson, you should be able to
- find the equation of a line that estimates nearly linear data by calculating the rate of change and estimating the vertical intercept of the line.
- use approximate linear models to interpolate and extrapolate.
PROBLEM SITUATION: LIFE EXPECTANCY AND SOCIAL SECURITY
This lesson will focus on the Social Security program. Social Security provides income for people who are elderly or disabled. The following information is from the Social Security Administration website:3
Social Security is financed through a dedicated payroll tax. Employers and employees each pay 6.2 percent of wages up to the taxable maximum of $160,200 (in 2023), while the self-employed pay 12.4 percent. When a person reaches “full retirement age,” he or she can claim a full retirement benefit. If someone retires before full retirement age, he or she receives a lower benefit.
(1) Use the following information from the Social Security Administration website to answer the questions below.4
Full retirement age is the age at which a person may first become entitled to full or unreduced retirement benefits. No matter what your full retirement age (also called "normal retirement age") is, you may start receiving benefits as early as age 62 or as late as age 70.
If You Retire Early
You can retire at any time between age 62 and full retirement age. However, if you start benefits early, your benefits are reduced a fraction of a percent for each month before your full retirement age.
The chart below lists age 62 reduction amounts and includes examples based on an estimated monthly benefit of $1000 at full retirement age.
Full Retirement and Age 62 Benefit By Year Of Birth
| Year of Birth | Full (normal) Retirement Age | Months between age 62 and full retirement age | A $1000 retirement benefit would be reduced to | The retirement benefit is reduced by | A $500 spouse’s benefit would be reduced to | The spouse’s benefit is reduced by | |||||||||
| 1937 or earlier | 65 | 36 | $800 | 20.00% | $375 | 25.00% | |||||||||
| 1938 | 65 and 2 months | 38 | $791 | 20.83% | $370 | 25.83% | |||||||||
| 1939 | 65 and 4 months | 40 | $783 | 21.67% | $366 | 26.67% | |||||||||
| 1940 | 65 and 6 months | 42 | $775 | 22.50% | $362 | 27.50% | |||||||||
| 1941 | 65 and 8 months | 44 | $766 | 23.33% | $358 | 28.33% | |||||||||
| 1942 | 65 and 10 months | 46 | $758 | 24.17% | $354 | 29.17% | |||||||||
| 1943-1954 | 66 | 48 | $750 | 25.00% | $350 | 30.00% | |||||||||
| 1955 | 66 and 2 months | 50 | $741 | 25.83% | $345 | 30.83% | |||||||||
| 1956 | 66 and 4 months | 52 | $733 | 26.67% | $341 | 31.67% | |||||||||
| 1957 | 66 and 6 months | 54 | $725 | 27.50% | $337 | 32.50% | |||||||||
| 1958 | 66 and 8 months | 56 | $716 | 28.33% | $333 | 33.33% | |||||||||
| 1959 | 66 and 10 months | 58 | $708 | 29.17% | $329 | 34.17% | |||||||||
| 1960 and later | 67 | 60 | $700 | 30.00% | $325 | 35.00% |
(a) Who can collect full retirement benefits at age 65?
(b) At what age can someone born in 1950 collect full benefits?
(c) What is the youngest age at which you can begin collecting reduced benefits?
One proposal to reform Social Security is to increase the retirement age. This, however, raises concerns about fairness. No matter what the retirement age is, some people will pay into Social Security but die before retirement and never receive a benefit. This happens to more people when the retirement age is increased. In this lesson, you will examine the effects of raising the retirement age to 75. Specifically, you will answer the question of whether this change would have a greater impact on some groups than others.
To explore this question, you will use life expectancy data from the Centers for Disease Control. Real data rarely fall on a straight line, but sometimes data show a definite trend. If the trend is close to linear, the data can be approximated by a linear model. This means that a linear model gives good estimates of what the data will be if the trend continues. A model can also be used to estimate values between data points. In this lesson, you will learn to create linear models from data.
The following data show the life expectancy of African American males in the United States at birth.
(2) Find a linear model to approximate these data, letting L be the life expectancy at birth and y the year of birth measured in years after 1900.
(3) In what year will African American male babies have a life expectancy of 75 years? In what year will these male babies be eligible to begin drawing Social Security if the retirement age is raised to 75?
Babies born in __________ will have an average lifespan of 75 years.
These babies will start drawing Social Security in: __________
(4) The last part of this lesson uses data about the life expectancy of a different population of Americans: white females.
Find a linear model to approximate your dataset, letting L be the life expectancy at birth and y the year of birth measured in years after 1900. Be prepared to develop the line you feel best represents the data graphically and to discuss how you found the equation of this line.
(5) When does your model predict the population will first have a life expectancy of 75 years at birth? In what year will this group begin collecting Social Security if the retirement age is raised to 75?
(6) When does your model predict the population will first have a life expectancy of 90 years at birth?
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3 https://www.ssa.gov/news/press/factsheets/HowAreSocialSecurity.htm
4 http://www.socialsecurity.gov/retire2/agereduction.htm#chart