Allegany College

Pre-Calculus 119 Final Project

By: Will Spencer

Instructor: Dr. Mark Shore

 

 

 

Project Title: Life Expectancy of White males versus Black Males

 

Data Source: National Center for Health Services

http://www.cdc.gov/nchs

Data set found at http://www.cdc.gov/nchs/fastats/pdf/47_28t12.pdf

 

Objective: To determine if there is a continual increase of life expectancy, and to see if there is an overlap of life expectancy of Black males versus white males.

 

Independent Variable: Years (T=0=1900 in decades. T=10=1910)

 

Dependant Measure 1: Life expectancy in White Males

 

Dependant Measure 2: Life expectancy in Black Males

 

Data Points for Life Expectancy for White Males:

 

Independent Variable

(Years)

Dependant Variable 1

(Life Expectancy of White Males)

 

 

1910

48.6

1920

54.4

1930

59.7

1940

62.1

1950

66.5

1960

67.4

1970

68.0

1980

70.7

1990

72.7

 

 

 

Interpolation: Finding the curve of best fit within the range of the data.

 

The following chart shows the model with the highest correlation coefficient based on the data from 1910 to 1990.

 

Equation Type:            Coefficient:          Model:

 

Linear                         R2= 0.9283          y= 0.2787x + 49.411

 

Quadratic                   R2= 0.985            y= -0.003x2 + 0.5828x + 43.836

 

Exponential                R2= 0.8993          y= 50.042e0.0046x

 

Logarithmic               R2= 0.989             y= 10.952Ln(x) + 22.548

 

Power                          R2= 0.9926          y= 31.843x0.1826

 

Cubic                          R2= 0.9942          y= 6E-05x3 - 0.0116x2 + 0.9429x + 40.075

 

The Equation with the highest correlation coefficient is the Cubic equation, which will be used now.

 

This chart reflects the actual data, compared to the predicted data using the Cubic equation.

 

Year

Age in Years using the Actual Data

Age in Years using the Predicted Data with the Cubic Equation

 

 

 

1910

48.6

48.40

1920

54.4

54.75

1930

59.7

59.47

1940

62.1

62.89

1950

66.5

65.37

1960

67.4

67.23

1970

68.0

68.83

1980

70.7

70.51

1990

72.7

72.60

 

 

 

 

 

In the long run, what does your linear equation predict for your data? This equation is not good for extrapolation because it predicts a steady rise in life expectancy into infinity. It does not level out anywhere, and therefore is invalid.

 

In the long run, what does your quadratic equation predict for your data? This equation is not good for extrapolation because after the year 1995, there is a steady increase in the life expectancy, and since we are alive today, it must be invalid.

 

In the long run, what does your cubic equation predict for your data? In the long run, the cubic equation is not good. It predicts that the life expectancy will continue to rise until infinity. This is not good for extrapolation. However, this equation is perfect for interpolation.

 

In the long run, what does your forth power equation predict for your data? This equation is not good for extrapolation because it starts with the life expectancy at infinity, and coming down at the year 1862 then increasing again, only to finally rise into infinity.

 

In the long run, what does your exponential equation predict for your data? This equation is good for interpolation, but like some of the others, the life expectancy continues to raise to infinity.

 

In the long run, what does your logarithmic equation predict for your data? This equation is okay for interpolation, and is a little better for extrapolation. The life expectancy slowly rises into infinity, but after an infinite amount of years.

 

In the long run, what does your power equation predict for your data? This equation is also decent for interpolation, but in the long run, slowly increases into infinity. However, it is the best for extrapolation because of the slow rate of the increase into infinity. This makes sense because more than likely, technology will increase at about the same rate, making longer life expectancies possible.

 

In your opinion, which equation is best for extrapolation? Why? Probably the best equation for extrapolation would have to be the power equation. It still raises the life expectancy into infinity, but it does it at a much slower rate than any of the other equations.

 

The power equation will be used to predict some extreme values for the future.

 

Year

Predicted Value using the Power equation

 

 

2050 (150)

79.51

3000 (200)

83.80

3500 (250)

87.28

4000 (300)

90.24

5000 (400)

95.11

 

 

Data Points for Life Expectancy for Black Males:

 

Independent Variable:

(Years)

Dependant Variable 2

(Life Expectancy, Black Males)

 

 

1910

33.8

1920

45.5

1930

47.3

1940

51.5

1950

59.1

1960

61.1

1970

60.0

1980

63.8

1990

64.5

 

 

 

Interpolation: Finding the curve that best fits within the range of the data.

 

 

The following chart shows the model with the highest correlation coefficient based on the data from 1910 to 1990.

 

 

 

Equation Type:            Coefficient:          Model:

 

Linear                         R2= 0.887            y= 0.3545x + 36.342

 

Quadratic                   R2= 0.9681          y= -0.0047x2 + 0.8276x + 27.669

 

Exponential                R2= 0.8349          y= 37.257e0.0071x

 

Logarithmic               R2= 0.9721          y= 14.131Ln(x) + 1.4296

 

Power                          R2= 0.9683         y= 18.008x0.2902

 

Cubic                          R2= 0.9711          y= 4E-05x3 - 0.0111x2 + 1.0946x + 24.88

 

 

 

The Equation with the highest correlation coefficient is the Logarithmic equation, which will be used now.

 

This chart reflects the actual data, compared to the predicted data using the Logarithmic equation.

 

Year

Age in Years

(Actual Data)

Age in Years with Predicted data from Logarithmic equation

 

 

 

1910

33.8

33.96

1920

45.5

43.76

1930

47.3

49.49

1940

51.5

53.55

1950

59.1

56.70

1960

61.1

59.28

1970

60.0

61.46

1980

63.8

63.35

1990

64.5

65.01

 

 

 

 

 

In the long run, what does your linear equation predict for your data? This equation is not good for extrapolation because it increases at a constant rate, and will increase until infinity.

 

In the long run, what does your quadratic equation predict for your data? This equation may be good for interpolation, but not extrapolation. The life expectancy increases until the year 1987, then it begins to decrease to below zero, which is impossible. And since we are alive today, 1987 would not be a valid ending date.

 

In the long run, what does your forth power equation predict for your data? This equation is not good for either interpolation or extrapolation. It begins with the life expectancy from previous years to be infinity, then in decreases and goes up and down again. It finally ends by going into infinity. This is not good for extrapolation.

 

In the long run, what does your cubic equation predict for your data? This equation is excellent for interpolation, but not for extrapolation. After 100 years, the life expectancy begins to increase too much making unrealistic values.

 

In the long run, what does your exponential equation predict for your data? This equation is good for interpolation, but much like the cubic equation, after 100 years the data begins to be unrealistic.

 

In the long run, what does your logarithmic equation predict for your data? This equation is not only good for interpolation, but it is also good for extrapolation. The increase of life expectancy is slow enough to give realistic values. For example, in the year 5000, the life expectancy is only 86 years old. This is great for extrapolation.

 

In the long run, what does your power equation predict for your data? This equation reacts similar to the logarithmic equation, but in the long run the numbers are not as realistic as the logarithmic equation, making it not the best for extrapolation.

 

In your opinion, which equation is best for extrapolation? Why? The best equation for extrapolation would have to be the logarithmic equation. In the long run its numbers are very realistic. A growth in technology would result in a longer life expectancy as the years go on, but it would have to be over many years. The logarithmic equation shows this beautifully.

 

The logarithmic equation will be used to predict some extreme values for the future.

 

Year

Predicted Value using the Logarithmic equation

 

 

2050 (150)

72.23

3000 (200)

76.29

3500 (250)

79.45

4000 (300)

82.02

5000 (400)

86.09

 

 

 

Equation that is best for extrapolation in Dependant Measure 1 (Life Expectancy of White Males):

 

Power                          R2= 0.9926          y= 31.843x0.1826

 

Equation that is best for extrapolation in Dependant Measure 2 (Life Expectancy of Black Males):

 

Logarithmic               R2= 0.9721          y= 14.131Ln(x) + 1.4296

 

There is no intersection point on this data, which means that the average life expectancy for White Males have and according to this data, always will be higher than that of Black Males.

 

Project Conclusions:

 

This data shows that the average life expectancy of white males have always been higher than that of black males. There are probably many reasons for this: sickle cell disease, segregation, and sadly, racism are all causes for a black males life expectancy to be shorter. However, with a steady increase in technology both white and black males have seen a steady increase in life expectancy and will continue to do so until a certain point.

 

 

Allegany College