Descriptive Statistics and Regression Analysis for Remuneration in Desklib
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This article provides a detailed analysis of remuneration in Desklib using descriptive statistics and regression analysis. It includes the mean, standard deviation, and range of remuneration, rank, and student number. It also discusses the estimated regression models and their coefficients.
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Running Head: MAE256 T1 MAE256 T1 Name of the Student Name of the University Course ID
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2MAE256 T1 Answer i Descriptive Statistics remuneration Mean 801.270270 3 Standard Error 27.6295650 7 Median805 Mode895 Standard Deviation 168.064083 1 Sample Variance 28245.5360 4 Kurtosis - 0.76314265 7 Skewness - 0.11940181 9 Range650 Minimum445 Maximum1095 Sum29647 Count37 rank Mean 336.810 8 Standard Error 23.2879 2 Median376 Mode450 Standard Deviation 141.654 9 Sample Variance20066.1 Kurtosis-0.32679 Skewness-1.01294 Range416
3MAE256 T1 Minimum34 Maximum450 Sum12462 Count37 studnum Mean 38410.2702 7 Standard Error 6023.83362 1 Median29214 Mode26704 Standard Deviation 36641.5494 4 Sample Variance1342603145 Kurtosis 27.0045643 8 Skewness 4.85680684 6 Range230347 Minimum9899 Maximum240246 Sum1421180 Count37 The mean remuneration of the states is obtained as 801. This implies that the average remuneration of Vice Chancellors’ in different states of Australia is 801. The mean of university ranks is 337.This indicates that average rank of universities obtained from Time Higher education is 337. The mean of studnum is 38410. That means there are 38410 number of students enrolled in the University on an average. The primary measures of dispersion include standard deviation, variance and range. The range of remuneration is 650. The remuneration to Vice Chancellors lies between 1095 and 445. The standard deviation of remuneration of 168.064.SD shows the divergence of data points
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4MAE256 T1 from the mean value. SD is less than average implying the coefficient of variation is less than 1. This means the distribution of remuneration is not much volatile. The range of rand and student number are 416 and 230347 respectively.The standard deviations of rank and student numbers are 141.6549 and 36641.54944 respectively. This indicates the distribution of rank and student number are dispersed widely from mean. Answer ii The estimated regression model of remuneration is remuneration=β0+β1rank+u The regression result is obtained as Regression Statistics Multiple R0.5788 R Square0.3350 Adjusted R Square0.3160 Standard Error138.9987 Observations37 ANOVA dfSSMSFSignificanceF Regression1340616.998340616.99817.6300.000 Residual35676222.29919320.637 Total361016839.297 Coefficient s Standard ErrortStatP-value Lower 95% Upper 95% t- critical Intercept1032.549459.634417.31460.0000911.48501153.61372.0281 rank-0.68670.1635-4.19880.0002-1.0187-0.35472.0281 From the regression result the estimated equation of remuneration is
5MAE256 T1 remuneration(^R)=1032.5494−(0.6867×rank) The coefficient of rank is -0.6867. This implies with 1 percent increase in rank remuneration decline by 0.69 percent. The estimated t value of the coefficient is (-0.6867/0.1635) = -4.2.The critical t value is 2.8. As the absolute value of computed t is greater than the critical t value, the null hypothesis no significant relation between rank and remuneration is rejected. The result is further supported by p value. P value of the estimate is 0.0002. The p value is less than the significance value of 0.05. Therefore, the variable rank is significant at 5% level of significance. Answer iii A model of remuneration is to be estimated using log-log specification which is given as follows log(remuneration)=β0+β1log(rank)+u The obtained regression result is given below Regression Statistics Multiple R0.5334 R Square0.2846 Adjusted R Square0.2641 Standard Error0.1899 Observations37 ANOVA dfSSMSFSignificanceF Regression10.500.5013.920.00 Residual351.260.04 Total361.76 Coefficient s Standard ErrortStatP-value Lower 95% Upper 95% t critical Intercept7.59520.251730.17290.00007.08428.10632.0281 log(rank)-0.16500.0442-3.73100.0007-0.2547-0.07522.0281
6MAE256 T1 log(remuneration)¿ The coefficient of log (rank) is -0.1650. The implication is that with every 1% increase in log(rank), log(remuneration) falls by 0.17%. The computed t value is (-0.1650/0.0042) =-3.7330. The absolute value of computed t is greater than critical value implying rejection of null hypothesis of no significant relation between log(rank) and log (remuneration). P value for the coefficient is 0.0007. The p value less than significant value of 0.05 which supports the result obtained from critical t value approach. The variable log(rank) therefore is a negative significant determinant of log (remuneration). Higher university rank is an indicator of poor performance. This therefore likely to have an adverse effect on remuneration. The sign of the estimated co- efficient thus matches with the expectation. Answer iv The next mode is t estimate to relate remuneration with rank and number of students remuneration=β0+β1rank+β2studnum+u Regression Statistics Multiple R0.7217 R Square0.5208 Adjusted R Square0.4926 Standard Error119.7109 Observations37 ANOVA
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7MAE256 T1 dfSSMSFSignificanceF Regression2529595.269264797.63418.4780.000 Residual34487244.02814330.707 Total361016839.297 CoefficientsStandardErrortStatP-valueLower95%Upp Intercept950.189656.144216.92410.0000836.09081 rank-0.66780.1409-4.73800.0000-0.9542 studnum0.00200.00053.63140.00090.0009 Estimated regression equation remuneration(^R1)=950.1896−0.6678rank+0.0020studnum Goodness of the fit of a regression model is determined from the value of R square. The R square value indicates how close the data set are fitted to the estimated regression model. For the fitted model in ii), the R square value is obtained as 0.34. This means rank can explain only 34% variation in remuneration. This is not a good fitted model. For regression model including more than one independent variable the adjusted R square is used for this purpose. After including student number as an independent variable along with rank R square value increases to 0.50. Now, the independent variables can explain 50% variation in remuneration. The second model is thus fitted well as compared to that in ii. Answer v The log-log model that is to be estimated is log(remuneration)=β0+β1log(rank)+β2log(studnum)+u
8MAE256 T1 The regression result is obtained as Regression Statistics Multiple R0.6454 R Square0.4165 Adjusted R Square0.3822 Standard Error0.1740 Observations37 ANOVA dfSSMSFSignificanceF Regression20.7340.36712.1360.000 Residual341.0290.030 Total361.763 Coefficient s Standard ErrortStatP-value Lower 95% Upper 95% Critical t Intercept5.91470.64849.12190.00004.59707.23242.7195 log(rank)-0.13810.0417-3.31470.0022-0.2227-0.05342.7195 log(studnum)0.14750.05322.77320.00890.03940.25562.7195 The estimated regression equation is log(remuneration)(^R1)=5.9147−0.1381log(rank)+0.1475log(studnum) The elasticity of remuneration with respect to student number is 0.1475. The computed t value of for studnum is (0.1475/0.0532) = 2.77. The critical t value at 1% level of significance is 2.7195. The computed t value thus falls in the rejection region. The rejection of null hypothesis of no significant relation a statistically significant relation exists between remuneration and student number. The p value of the concerned variable is 0.0089. The p value is less than significance value of 0.01. This again indicates the variable is statistically significant.
9MAE256 T1 Answer vi The coefficient of rank is -0.1381. Negative sign of the coefficient implies a negative relation between the two variables. The computed t value is (-0.1381/0.0417)= -3.31. The computed t value in absolute term is greater than the critical t value of 2.72 at 1% level of significance. The variable therefore is statistically significant. The corresponding p value is 0.0022. The p value is less than the significance level implying studnum is statistically significant at 1% level of significance. Answer vii The new proposed model is log(remuneration)=β0+β1log(rank)+β2log(studnum)+β3grademp+β4gradstudy+u Regression Statistics Multiple R0.7197 R Square0.5179 Adjusted R Square0.4576 Standard Error0.1630 Observations37 ANOVA dfSSMSFSignificanceF Regression40.9130.2288.5940.000 Residual320.8500.027 Total361.763 Coefficient s Standard ErrortStat P- value Lower 95% Upper 95% Criticalt (1%) Intercept4.70370.92555.08240.00002.81856.58892.7195 log(rank)-0.04030.0546 - 0.73840.4657-0.15150.07092.7195 log(studnum )0.13080.05052.59110.01430.02800.23362.7195
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10MAE256 T1 grademp0.00740.00621.18900.2432-0.00530.02012.7195 gradstudy0.01350.00522.58900.01440.00290.02422.7195 Estimated regression equation log(remuneration)(^R1)=4.7037−0.0403log(rank)+0.1308log(studnum)+0.0074grademp+0.0135grdestudy Computed t value for each of the independent variable is log(rank):t=−0.0403 0.0546=−0.7384 log(studnum):t=0.1308 0.0505=2.5911 grademp:t=0.0074 0.0062=1.1890 gradestudy:t=0.0135 0.0052=2.5890 The computed t values for all the four independent variables are less than the critical t value. This indicates all the variables are statistically insignificant. The result obtained from critical t value approach is supported from the p value where all the p values are less than significance value of 0.01. In order to test overall significance F test needs to be conducted The hypotheses of the F test are Null hypothesis (H0):β1=β2=β3=β4=0 Alternative hypothesis (H1); at least one ofβs is not zero.
11MAE256 T1 F= RSS K SSE [n−(k+1)] =Meanregrssion∑ofsquare Meansquarederror=MSR MSEF4,32 From the regression result the computed value of F is obtained as F= 0.913 4 0.850 32 =0.228 0.027=8.594 The tabulated value ofF4,32at 5% level of significance is 2.668. The computed F value is greater thanthetabulatedFvalue.Therefore,thenullhypothesisstatingthemodelisoverall insignificantisrejected.Themodelthereforehasanoverallsignificance.Thepvalue corresponding to the F statistics is 0.000. As the p value is less than the significance value of 0.05, again suggesting rejection of the null hypothesis. Hence, is can be concluded that the model is overall significant at 5% level of significance. Answer viii The critical F value at 1% level of significance is 3.9694. This is again less than computed F value implying an overall significance of the model at 1% level of significance. Answer ix In order to test whether Vice Chancellors of universities located in Victoria are paid a higher remuneration as compared to other state, a dummy variable called state dummy generated. The variable assumes a value 1 if the state is Victoria and a value of 0 otherwise. The regression equation to be estimated is log(remuneration)=β0+β1log(rank)+β2log(studnum)+β3(statedummy)+u
12MAE256 T1 The framed hypothesis is Null hypothesis (H0): The variable state dummy has no significant relation with remuneration Alternative Hypothesis (H1): The variable state dummy has a statistically significant relation with remuneration. The regression results are given below Regression Statistics Multiple R0.6502 R Square0.4227 Adjusted R Square0.3703 Standard Error0.1756 Observations37 ANOVA dfSSMSFSignificanceF Regression30.7450.2488.0550.000 Residual331.0180.031 Total361.763 Coefficient s Standard ErrortStat P- value Lower 95% Upper 95% Critical t Intercept5.89360.65568.98960.00004.55987.22752.0281 log(rank)-0.13680.0421-3.24930.0027-0.2225-0.05122.0281 log(studnum)0.14790.05372.75410.00950.03860.25722.0281 State Dummy0.04010.06740.59520.5558-0.09700.17722.0281 Computed t value for the state dummy is (0.0401/0.0674) = 0.5952. The computed t value is less than the critical t value of 2.0281 at 1% level of significance. As the t computed t values fall in the acceptance region, it therefore indicates acceptance of the null hypothesis. The state dummy therefore does not have any significant influence on remuneration. The result can
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13MAE256 T1 be further supported by observing the p value of 0.5558, which is greater than the significance level of 0.05. Therefore, it can be concluded that the Vice Chancellors of Victoria are not likely to receive a higher remuneration as compared to those in other states.
14MAE256 T1 Bibliography Carrasco, M., Chernozhukov, V., Gonçalves, S. and Renault, E., 2015. High dimensional problems in econometrics.JournalofEconometrics,2(186), pp.277-279. Henderson, D.J. and Parmeter, C.F., 2015.Appliednonparametriceconometrics. Cambridge University Press. Magnusson,L.M.,2016.EconometricsinaFormalScienceofEconomics:Theoryand Measurement of Economic Relations, by Bernt P. Stigum (MIT Press, Cambridge, 2015), pp. 392.EconomicRecord,92(298), pp.509-511. Mazodier,P.,2017.MrMalinvaudandEconometrics.AnnalsofEconomicsand Statistics/Annalesd'ÉconomieetdeStatistique, (125/126), pp.169-185. Molchanov, I. and Molinari, F., 2018.Randomsetsineconometrics(Vol. 60). Cambridge University Press. Wooldridge, J.M., Wadud, M. and Lye, J., 2016.IntroductoryEconometrics:AsiaPacific EditionwithStudentResourceAccessfor12Months. Cengage AU.
15MAE256 T1
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