Projections of the Number of Persons Living Alone by Gender

Purpose: The purpose of this study is to show the correlation between the calendar year and the number of males and females living alone. The study also makes projections for the number of men and women living alone in the future. The study concludes with any possible year in which the number of men and women living alone are equal.

Data on women living alone:

Year with t = 1 is 1995	Total number of women living alone age 15 or above (units in millions)
1	14.131
2	14.303
3	14.48
4	14.664
5	14.843

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

Linear equation of the form Y = at+b W = .1785x+13.949

Coefficient of determination (r²) = .9998624047

Quadratic Equation of the form Y=at²+bt+c W= .0015x² + .1695x+ 13.959

Coefficient of determination (r²) = 1

Exponential equation of the form Y=ae^bt W=13.956344e^.0123241

Coefficient of determination (r²) = 1

Logarithmic equation of the form Y= a+blnt W= 14.0713796 +.43114475lnx

Coefficient of determination (r²) = .9423525338

Power equation of the form Y= at^b W= 14.0743958x^.0298165

Coefficient of determination (r²) = .9455539225

The equation that gives the highest r² value is the quadratic equation.

W = .0015x² + .1695x + 13.959

Independent variable	Dependent variable	Predicted value from equation with the highest r^2 W = .0015x^2 + .1695x + 13.959
1	14.131	14.3
2	14.303	14.304
3	14.480	14.481
4	14.664	14.661
5	14.843	14.844

Extrapolation: Predicting values outside the range of the data.

In the long run, what does your linear equation predict for your data? The number of women living alone will continue to rise .1785 million each year. This equation is not good for extrapolation because the numbers of women that live alone keep getting higher. Unless there are no men in the world this equation is not realistic.

In the long run, what does your quadratic equation predict for your data? This equation is realistic within the data points. But once outside the numbers increase too quickly and one day there will be negative women living alone. Not good for extrapolation

In the long run, what does your cubic equation predict for your data? The equation is extremely accurate within the data points. It also shows a slight increase but then begins to drop drastically. By the year 2030 the number of women that live alone will be –5.485 million, which is totally unrealistic. This is not good for extrapolation.

In the long run, what does your fourth power equation predict for your data? This equation was good for interpolation and it even showed a realistic increase in the year 2003. But by the year 2015 the number of women living alone was –34.067 million. This is not good for extrapolation.

In the long run, what does your exponential equation predict for your data? This equation predicts rapid growth. Extrapolation gives realistic values to a certain point. But with further extrapolation the numbers become unrealistically large.

In the long run, what does your logarithmic equation predict for your data? This equation predicts a slower increase. This is good for both interpolation and extrapolation. It shows a slower increase over a period of time. Which is what you would expect.

In the long run, what does your power equation predict for your data? This equation is also good for both interpolation and extrapolation. It also shows a slow increase over a period of time.

In your opinion, which equation is the best for extrapolation. Why? Both the logarithmic and power equations are good for extrapolation because they show a slow growth over a period of time. Which fits reality.

Although both the power and the logarithmic fit extrapolation I will use the logarithmic equation W= 14.0713796 +.43114475lnx for extrapolation on the table below.

Independent variable	Predicted value from the equation you believe is the best for extrapolation W= 14.0713796 +.43114475lnx
10	15.064
40	15.661
60	15.836
100	16.056
200	16.255

Data on men living alone:

Year with T=1 is 1995	Total number of men living alone 15 or above (units of millions)
1	10.172
2	10.381
3	10.586
4	10.788
5	10.988

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

Linear equation of the form Y = at + b M = .2039x + 9.99713

Coefficient of determination (r²) = .9999232615

Quadratic equation of the form Y = at² + bt = c M = -.00015x² + .2129x + 9.9608

Coefficient of determination (r²) = 1

Exponential equation of the form Y = ae^bt M = 9.98457309e^.0192787

Coefficient of determination (r²) = .99959338113

Logarithmic equation of the form Y = a + blnt M = 10.1092601 + .49476834lnx

Coefficient of determination (r²) = .9510653436

Power equation of the form Y = at^b M = 10.1144774x^.04690296

Coefficient of determination (r²) = .9558117627

The equation that gives the highest r² value is the quadratic equation.

M = -.00015x² + .2129x + 9.9608

Independent variable	Dependent variable	Predicted value from the equation with the highest r^2 M = -.00015x^2 + .2129x + 9.9608
1	10.172	10.172
2	10.381	10.38
3	10.586	10.586
4	10.788	10.788
5	10.988	10.987

Extrapolation: Predicting values outside the range of the data.

In the long run, what does your linear equation predict for your data? The number of men living alone increases .2039 million each year. This equation is not good for extrapolation unless the number of the male population grows exponentially to the female population.

In the long run, what does your quadratic equation predict for your data? This equation is good for interpolation. But once outside the data the number of men living alone increases too rapidly. Eventually the number of men who live alone will be negative. This is not good for extrapolation.

In the long run, what does your cubic equation predict for your data? This equation is realistic inside the data. Since it is a cubic equation the graph shows two humps so the graph will not change direction it will continue to increase. For example in the year 2045, the number of men living alone will be 34.134 million.

In the long run, what does your fourth power equation predict for you data? This equation predicts that the number of men living alone will continuously increase. For interpolation the equation is good. But with extrapolation, the results may be unrealistic. For example in the year 2094, the number of men will be 168.3 million, which is impossible.

In the long run, what does your exponential equation predict for your data? This equation fits within the data points. Extrapolation gives realistic numbers. In the year 2094, the number of men will be 68.64 million, which is possible depending on population growth. However, with further extrapolation the numbers are unrealistic.

In the long run, what does your logarithmic equation predict for your data? This equation is good for both interpolation and extrapolation. It shows a slower increase over a longer period of time, which is what you would expect.

In the long run, what does your power equation predict for your data? This equation is good for extrapolation, because like the logarithmic, it shows a slower increase over a long period of time.

In your opinion, which equation is best for extrapolation? Why? In my opinion both the logarithmic and power equations are good, because they both show a slow increase over a long period of time. These equations are better than the others because all of the other equations show rapid increases or decreases which is impossible for the data.

Although both logarithmic and power equations are good for extrapolation, I will use logarithmic equation M = 10.1092601 + .49476834lnx.

Independent variable	Predicted value from the equation you believe is best for extrapolation M = 10.1092601 + .49476834lnx
10	11.248
40	11.934
60	12.135
100	12.387
200	12.730

Data on men and women living alone:

Year with t = 1 is 1995	Total number of women living alone age 15 or above (units in millions)	Year with T=1 is 1995	Total number of men living alone 15 or above (units of millions)
1	14.131	1	10.172
2	14.303	2	10.381
3	14.48	3	10.586
4	14.664	4	10.788
5	14.843	5	10.988

The equation of best fit I believe is best for extrapolation for the independent variable and the first dependent measure is the logarithmic equation.

W = 14.0713796 + .43114475lnx

The equation of best fit I believe is best for extrapolation for the independent variable and the second dependent measure is the logarithmic equation.

M = 10.1092601 + .497476834lnx

Explain what the points of intersection means for the two dependent measures.

In the year 9176, there will be exactly the same number of males living alone as there are females living alone.

Conclusion: The number of women living alone has stayed higher than the number of men throughout the past five years. But the number of men living alone seems to be growing at a much faster rate. With extrapolation the number of men living alone will eventually surpass the number of women living alone.