Question

Page 11 of 12 SOCIAL SECURITY BENEFICIARIES The table shown below gives the numbers of Social Security beneficiaries' in millions) for selected years from 1950 to 2000, with projections through 2030 Year 1950 1960 1970 1980 1990 2000 2010 20202 2030 Beneficiaries 2.9 14.3 25.2 35.1 44.8 (millions) Source: 2000 Social Security Trustees Record) (34.) Draw a scatter diagram that models the data shown in the table above. Find the cubic regression equation that models the data, with x equal to the number of years after 1950 and B(x) equal to the number of millions of beneficiaries. Report the model with four decimal places B Use the increasing and Decreasing Test for derivatives to indicate the intervals where the function is increasing and decreasing. Please show your analysis. Use the cubic regression model developed in above to predict the years with extremum (both maximum and minimum) number of beneficiaries.

Answer 1

Let the cubic estimation be

$y=ax^{3}+bx^{2}+cx+d$

where x is numbers of years after 1950 and y is number of beneficiaries. The scatter plot on excel is obtained as below.

0 10 20 30 40 50 60 70 80 90

We shall do regression analysis on excel Solver. For this, we initialize the parameters a, b, c, d of the regression equation and compute the estimated values, then find the sum of squared estimate of errors (SSE)

7.84 20 Years since 1950 Beneficiaries Estimated Value (SSE Calculation 02.9 0.1 14.3 2.2|| 146.41 25.2 6.9 334.89 30 35.1 14.8 412.09 40 39.5 26.5 169 44.8 42.6 4.84 60 53.3 108.16 70 68.8 90.41 466.56 80 8 2.7 123.3 1648.36 3298.151 63.7 Regression Parameters 0.0001 0.01 0.1 c 0.1

The SSE is right now not optimized. We shall let Solver do the work

Solver Parameters B Set Objective: $E$12 2.9 TO: O Max OMO O value of: 0 2.2 By Changing Variable Cells: $C$15:$C$18 Years since 1950 Beneficiaries Estimated Value SSE Calculation 0.1 7.84 14.3 146.41 25.2 6.9 334.89 35.1 14.8 412.09 39.5 26.5 169 44.8 42.6 4.84 53.3 63.7 108.16 68.8 90.4 466.56 82.7 123.3 1648.36 3298.15 Subject to the Constraints: Add Change Delete Regression Parameters 0.0001 0.01 Reset All Load/Save Make Unconstrained variables Non-Negative Select a Solving Method: GRG Nonlinear Options Solving Method Select the GRG Nonlinear engine for Solver Problems that are smooth nonlinear. Select the LP Simplex engine for linear Solver Problems, and select the Evolutionary engine for Solver problems that are non-smooth. T Help Sheet1 solve Sheet2 KE Ready Sheet3 Close

Hit Solve to obtain

Solver Results 2.9 Solver has converged to the current solution. All Constraints are satisfied. Years since 1950 Beneficiaries Estimated Value SSE Calculation 2.198390262 0.492256225 14.3 15.81601587| 2.298304111 25.2 25.581231091 0.145337148 35.11 32.92405808 4.734723251 39.5 39.27451895 0.050841705 44.8 46.06263584 1.594249268 53.3 54.71843089 2.011946191 68.8 66.67192623 4.528697976 82. 7 8 3.35314399 0.426597071 16.28295295 Reports Answer Sensitivity Limits O Keep Solver Solution O Restore Original Values Return to Solver Parameters Dialog Outline Reports OK Cancel Save Scenario... Solver has converged to the current solution. All Constraints are satisfied. Regression Parameters 0.000238337 -0.026412163 1.602050484 2.198390262 Solver has performed 5 iterations for which the objective did not move significantly. Try a smaller convergence setting, or a different starting point.

Hence, the exact cubic equation is

$y=0.000238337~x^{3}-0.026412163~x^{2}+1.602050484~x+2.198390262$

which can be expressed to desired accuracy. Its derivative is

$\frac{dy}{dx}=0.000715011~x^{2}-0.052824325~x+1.602050484$

The roots of this quadratic equations are not real. Hence, the function is always increasing as its slope is strictly positive. There is no extremum. It is quite natural to expect so, since the values given are always increasing. To verify further, we can plot the points and function together on Desmos as below

4 (80, 82.7) + y= 0.000238337,3 – 0.026412163x² + (0,2.9)(10,14.3),(20,25.2),(30,35.1). L 0 (70, 68.8) Label: (40, 39.5) (30, 35.1) (60, 53.3) (50, 44.8) (20, 25.2) (10, 14.3) (0, 2.9) 80