Effect of Age and Degree on Fulltime-Employed Adults Income - Essay Example

IBD BA1 Assignment QUANTITATIVE ANALYSIS Issued: 03 NOVEMBER Hand in by: 01 DECEMBER Effect of Age and Degree on Fulltime-Employed Adult’s Income A stratified random sample of adults who were in full time employment were selected. The stratifying criterion was the type of degree held, either Arts or Science. For each person selected, their income and age was noted. This paper will investigate how the income depends on type of degree and age using simple regression analysis. a) Scattergraph Scattergraph is used initially to investigate the possible relationship between two variables. Scattergraph shows if there exists a linear relationship or not between two variables. If the data points cluster around a straight line with increasing or decreasing slope, there exists a linear (positive or negative) relationship between two variables. Figure 1 shows the scattergraph of fulltime-employed adults against their age. Figure 1: Scattergraph of Income versus Age Figure 2 shows the scattergraph of fulltime-employed adults against their degree (Arts or Science). Figure 2: Scattergraph of Income versus Degree b) Scattergraph Interpretation There appears very strong linear relationship between income and age, as the data points lie (cluster) around a straight line. There appears very week (or no) linear relationship between income and degree. However, the scattergraph is inconclusive, as there are two values for degree Arts (0) and Science (1), only. c) Next Step based on the Scattergraph Interpretation of (b) The next step is to estimate a regression analysis to ensure the best fit. In other words, perform simple linear regression analysis to investigate how the income depends on type of degree and age. d) Fitted Model and “Goodness of Fit Measure”, from First Principles Regression Model 1: The explanatory (independent) variable is fulltime-employed adult’s age (x) and response (dependent) variable is income (y). According to Doane and Seward (2007), the fitted linear regression model is given by {Income = b0 + b1(Age)} Where, Slope, Intercept, Table 1 shows the calculation slope, b1 and intercept, b0 for Income vs. Age. Table 1: Calculation of slope and intercept (Income vs. Age) Age (x) Income(y) 25 24.10 -16.1 -12.44 259.21 154.63 200.20 26 25.00 -15.1 -11.54 228.01 133.06 174.18 27 26.00 -14.1 -10.54 198.81 110.99 148.54 28 26.60 -13.1 -9.94 171.61 98.70 130.15 30 28.20 -11.1 -8.34 123.21 69.47 92.52 32 29.30 -9.1 -7.24 82.81 52.35 65.84 35 30.70 -6.1 -5.84 37.21 34.05 35.59 37 34.20 -4.1 -2.33 16.81 5.45 9.57 38 34.90 -3.1 -1.64 9.61 2.67 5.07 39 34.10 -2.1 -2.44 4.41 5.93 5.11 41 35.50 -0.1 -1.04 0.01 1.07 0.10 43 36.40 1.9 -0.13 3.61 0.02 -0.26 44 40.00 2.9 3.47 8.41 12.01 10.05 46 41.10 4.9 4.57 24.01 20.84 22.37 48 40.60 6.9 4.07 47.61 16.52 28.05 50 45.00 8.9 8.47 79.21 71.66 75.34 55 49.00 13.9 12.47 193.21 155.38 173.26 57 46.70 15.9 10.17 252.81 103.33 161.62 60 53.50 18.9 16.97 357.21 287.81 320.64 61 49.80 19.9 13.27 396.01 175.96 263.97 = 822 = 730.70     = 2493.80 = 1511.89 = 1921.93 The mean age and income is calculated as The slope and intercept for the fitted regression model are Slope, Intercept, Therefore, the fitted regression model to predict income based on age is given by ‘Income = 4.860 + 0.771(Age)’. Assessing Fit: Goodness of Fit Measure The total variation in Y (denoted SST) around its mean is given by (total sum of squares) The explained variation (denoted SSR) in Y is given by (Regression sum of squares, explained) The unexplained variation in Y (denoted SSE) is given by (Error sum of squares, unexplained) Table 2 shows the calculation of sum of squares for Income vs. Age. The SST, SSR and SSE are 1511.89, 1481.20 and 30.69, respectively. The coefficient of determination, R2 is a measure of relative fit based on a comparison of SSR and SST, and is given by or In a bivariate regression, R2 is a measure of the correlation coefficient r. Correlation coefficient value near 0 indicates there is little (or no) association between the two variables and a value near 1 indicates a strong association between the two variables (Lind, Marchal and Wathen, 2009). Using correlation coefficient, R2 can be calculated as below Table 2: Calculation of Sums of Squares (Income vs. Age) Age (x) Income(y) Residual 25 24.10 24.127 -0.0270 0.0007 153.9585 154.6292 26 25.00 24.898 0.1023 0.0105 135.4272 133.0562 27 26.00 25.668 0.3316 0.1100 118.0837 110.9862 28 26.60 26.439 0.1610 0.0259 101.9282 98.7042 30 28.20 27.980 0.2196 0.0482 73.1809 69.4722 32 29.30 29.522 -0.2218 0.0492 49.1852 52.3452 35 30.70 31.834 -1.1338 1.2856 22.1010 34.0472 37 34.20 33.375 0.8248 0.6803 9.9843 5.4522 38 34.90 34.146 0.7541 0.5687 5.7079 2.6732 39 34.10 34.917 -0.8166 0.6668 2.6193 5.9292 41 35.50 36.458 -0.9579 0.9176 0.0059 1.0712 43 36.40 37.999 -1.5993 2.5578 2.1442 0.0182 44 40.00 38.770 1.2300 1.5129 4.9951 12.0062 46 41.10 40.311 0.7887 0.6220 14.2608 20.8392 48 40.60 41.853 -1.2527 1.5693 28.2781 16.5242 50 45.00 43.394 1.6059 2.5790 47.0470 71.6562 55 49.00 47.247 1.7525 3.0713 114.7576 155.3762 57 46.70 48.789 -2.0889 4.3634 150.1572 103.3272 60 53.50 51.101 2.3991 5.7556 212.1659 287.8112 61 49.80 51.872 -2.0716 4.2915 235.2112 175.9602 = 822 = 730.70 0.0000 SSE = 30.6862 SSR = 1481.1993 SST = 1511.8855 Regression Model 2: The explanatory variable is fulltime-employed adult’s degree (x) and response variable is income (y). 1 represents Science degree and 0 represents Arts degree. Let the fitted linear regression model is given by {Income = b0 + b1(Degree)} Table 3 shows the calculation slope, b1 and intercept, b0 for Income vs. Degree. Table 3: Calculation of slope and intercept (Income vs. Degree) Degree (x) Income(y) 0 24.10 -0.5 -12.44 0.25 154.63 6.22 1 25.00 0.5 -11.54 0.25 133.06 -5.77 0 26.00 -0.5 -10.54 0.25 110.99 5.27 1 26.60 0.5 -9.94 0.25 98.70 -4.97 1 28.20 0.5 -8.34 0.25 69.47 -4.17 0 29.30 -0.5 -7.24 0.25 52.35 3.62 1 30.70 0.5 -5.84 0.25 34.05 -2.92 0 34.20 -0.5 -2.33 0.25 5.45 1.17 0 34.90 -0.5 -1.64 0.25 2.67 0.82 1 34.10 0.5 -2.44 0.25 5.93 -1.22 1 35.50 0.5 -1.04 0.25 1.07 -0.52 1 36.40 0.5 -0.13 0.25 0.02 -0.07 0 40.00 -0.5 3.47 0.25 12.01 -1.73 0 41.10 -0.5 4.57 0.25 20.84 -2.28 1 40.60 0.5 4.07 0.25 16.52 2.03 0 45.00 -0.5 8.47 0.25 71.66 -4.23 0 49.00 -0.5 12.47 0.25 155.38 -6.23 1 46.70 0.5 10.17 0.25 103.33 5.08 0 53.50 -0.5 16.97 0.25 287.81 -8.48 1 49.80 0.5 13.27 0.25 175.96 6.63 = 10 = 730.70     = 5.00 = 1511.89 = -11.75 The mean degree and income is calculated as The slope and intercept for the fitted regression model are Slope, Intercept, Therefore, the fitted regression model to predict income based on degree is given by ‘Income = 37.71 – 2.35(Degree)’. Assessing Fit: Goodness of Fit Measure Table 4 shows the calculation of sum of squares. The SST, SSR and SSE are 1511.89, 27.61 and 1484.27, respectively. The coefficient of determination, R2 is In a bivariate regression, R2 is a measure of the correlation coefficient r. Table 4: Calculation of Sums of Squares for Income vs. Degree Degree (x) Income(y) Residual 0 24.10 37.710 -13.6100 185.2321 1.3806 154.6292 1 25.00 35.360 -10.3600 107.3296 1.3806 133.0562 0 26.00 37.710 -11.7100 137.1241 1.3806 110.9862 1 26.60 35.360 -8.7600 76.7376 1.3806 98.7042 1 28.20 35.360 -7.1600 51.2656 1.3806 69.4722 0 29.30 37.710 -8.4100 70.7281 1.3806 52.3452 1 30.70 35.360 -4.6600 21.7156 1.3806 34.0472 0 34.20 37.710 -3.5100 12.3201 1.3806 5.4522 0 34.90 37.710 -2.8100 7.8961 1.3806 2.6732 1 34.10 35.360 -1.2600 1.5876 1.3806 5.9292 1 35.50 35.360 0.1400 0.0196 1.3806 1.0712 1 36.40 35.360 1.0400 1.0816 1.3806 0.0182 0 40.00 37.710 2.2900 5.2441 1.3806 12.0062 0 41.10 37.710 3.3900 11.4921 1.3806 20.8392 1 40.60 35.360 5.2400 27.4576 1.3806 16.5242 0 45.00 37.710 7.2900 53.1441 1.3806 71.6562 0 49.00 37.710 11.2900 127.4641 1.3806 155.3762 1 46.70 35.360 11.3400 128.5956 1.3806 103.3272 0 53.50 37.710 15.7900 249.3241 1.3806 287.8112 1 49.80 35.360 14.4400 208.5136 1.3806 175.9602 = 10 = 730.70 0.0000 SSE = 1484.2730 SSR = 27.6125 SST = 1511.8855 e) Comparison of Fitted Models The regression model to predict fulltime-employed adult’s income based on their age is given by Income = 4.860 + 0.771(Age) (R2 = 0.9797) Where, income is in thousands of pounds (£000) and age is in years. The regression slope coefficient of 0.771 suggests that every year increase in fulltime-employed adult’s age increases income by about £771, on average. The regression intercept coefficient of 4.860 is not meaningful in the context of this problem because there will be no employee with 0 years age. The value of coefficient of determination, R2 is 0.9797. This suggests that fulltime-employed adult’s age explains about 97.97% variation in income. Only 2.03% variations in income remains unexplained. The regression model to predict fulltime-employed adult’s income based on their degree is given by Income = 37.71 – 2.35 (Degree) (R2 = 0.0183) Where, income is in thousands of pounds (£000) and Arts degree is represented by 0 and Science degree is represented by 1. The regression slope coefficient of -2.35 and intercept coefficient of 37.71 suggests that Science degree decreases fulltime-employed adult’s income by about £2,350 as compared to Arts degree. Arts fulltime employed adult’s income is £37,710, on average. The value of coefficient of determination, R2 is 0.0183. This suggests that fulltime-employed adult’s degree only explains about 1.83% variation in income and 98.17% variations in income remains unexplained. Thus, age is a good predictor of fulltime-employed adult’s income (R2 close to 1) In conclusion, fulltime-employed adult’s income is very strongly associated with their age and degree have very little or no effect. As employees age increases income also increases. References Doane, D.P. and Seward, L.E. (2007) Applied Statistics in Business and Economics, London: McGraw-Hill/Irwin. Lind, D.A., Marchal, W.G. and Wathen, S.A. (2009) Statistical Techniques in Business and Economics, 13th Edition, London: McGraw-Hill/Irwin. Appendix (for check) Simple Regression Analysis Using Excel (Income vs. Age) SUMMARY OUTPUT Regression Statistics Multiple R 0.98979967 R Square 0.979703386 Adjusted R Square 0.978575797 Standard Error 1.305674701 Observations 20 ANOVA   df SS MS F Significance F Regression 1 1481.199 1481.199 868.8475 1.09419E-16 Residual 18 30.68616 1.704786 Total 19 1511.886         Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Intercept 4.859916593 1.113553 4.364334 0.000374 2.520428787 7.199404399 Age(years) 0.770683295 0.026146 29.47622 1.09E-16 0.715752723 0.825613866 Simple Regression Analysis Using Excel (Income vs. Degree) SUMMARY OUTPUT Regression Statistics Multiple R 0.135142956 R Square 0.018263619 Adjusted R Square -0.036277292 Standard Error 9.080727455 Observations 20 ANOVA   df SS MS F Significance F Regression 1 27.6125 27.6125 0.334861 0.569981441 Residual 18 1484.273 82.45961 Total 19 1511.8855         Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Intercept 37.71 2.871578157 13.13215 1.17E-10 31.67703817 43.74296183 Degree -2.35 4.061024775 -0.57867 0.569981 -10.88189644 6.181896442 Read More