One important measure of workers' performance is their attendance record. A personnel manager at a large company with a serious absenteeism problem undertook a two-year study. She took a random sample of 450 employees and extracted information (see list below) from their employee records. Then she instituted a new employee Health and Wellness program to try to reduce absenteeism. At the end of one year, she went back to the records of the same 450 employees and recorded the number of days absent post-program. Here are some of the variables for which she collected data (the variables are labeled V1 through V6, for convenience). Assume V2, V4, V5, and V6 are normally distributed. ? Vl: Gender (1=male, 2=female) ? V2: Age (in years) ? V3: Employee status (1=full-time, 2=part-time, 3=casual) ? V4: Number of days absent in the twelve months prior to the program (0 to 200) ? V5: Employee performance evaluation (0 to 100) ? V6: Number of days absent in the twelve months after the program (0 to 200) For each of the following research questions, suggest which of the techniques we have studied in class could be used to address the question. A list of techniques to choose from is enumerated below. Select the number of the single most appropriate techniques from the drop-down list on the left of the research question. Note that some techniques may be used more than once, others not at all. Does age (V2). in years, ' have any effect on pre-program absenteeism (V4)?

One important measure of workers' performance is their attendance record. A personnel manager at a large company with a serious absenteeism problem undertook a two-year study. She took a random sample of 450 employees and extracted information (see list below) from their employee records. Then she instituted a new employee Health and Wellness program to try to reduce absenteeism. At the end of one year, she went back to the records of the same 450 employees and recorded the number of days absent post-program. Here are some of the variables for which she collected data (the variables are labeled V1 through V6, for convenience). Assume V2, V4, V5, and V6 are normally distributed. . V1: Gender (1=male, 2=female) . V2: Age (in years) . V3: Employee status (1=full-time, 2=part-time, 3=casual) . V4: Number of days absent in the twelve months prior to the program (0 to 200) . V5: Employee performance evaluation (0 to 100) . V6: Number of days absent in the twelve months after the program (0 to 200) For each of the following research questions, suggest which of the techniques we have studied in class could be used to address the question. A list of techniques to choose from is enumerated below. Select the number of the single most appropriate technique from the drop-down list on the left of the research question. Note that some techniques may be used more than once, others not at all. Does age (V2), in years, have any effect on pre- program absenteeism (V4)?1. {a} 1lr'lihat is the probability that the annual return will be between 6% and 14%? 2. {b} Suppose the politician earns more than 8% at the end of the year. 1y'uihat is the posterior probability that the investment 1was in Company B? 3. {c} Suppose the trustee inyested 30% ofthe politician's funds in Company A and ?{I}% in Company E. Find the expected annual return and the variance of this portfolio. Does age (V2), in years, have any effect on pre- program absenteeism (V4)? Do males and females (V1) differ with respect to pre- program absenteeism (V4)? Do all three levels of employment status (V3) 1. One-sample z-test of a proportion have the same average performance evaluation (V5)? 2. Two-sample z-test of two proportions Do males and females (V1) differ with respect to 3. One-sample t-test of a mean employment status (V3)? 4. Two-sample t-test of two Is the mean number of days independent means absent prior to the program (V4) different from 5. Matched pairs t-test of two the national average dependent means (available from Canada Page 1 Census studies)? 6. Chi-square test of independence Did the Health and Wellness program reduce 7. Simple linear regression the number of days absentDo males and females (V1) differ with respect to 3. One-sample t-test of a mean employment status (V3)? 4. Two-sample t-test of two Is the mean number of days independent means absent prior to the program (V4) different from 5. Matched pairs t-test of two the national average dependent means (available from Canada Census studies)? 6. Chi-square test of independence Did the Health and Wellness program reduce 7. Simple linear regression the number of days absent from pre-program (V4) to 8. Multiple regression post-program (V6)? 9. One-way analysis of variance Can pre-program absenteeism (V4) be explained by employee performance evaluation (V5) and age (V1)? Is there a disproportionate number of male employees in the sample (i.e., more than 50%) (V1)?Jelena Rusov Applying Regression Models to Predict PHD Shatera Business Results Danny ougurange & do, Belgrade Mirjana Misita In terms of modern business practice, business prediction rendi are Unventy al Degra crucially important for evaluation of future financial performance of a Faculty of Mechanical Engineer company. Planning and prediction procedures are rapertally important for Dragan D. Milanovic companies operating under uncertainty. This paper thou an couple of planning and prediction of baines results in insurance when calculating University of Belgrade premium trend by me of linear and nonlinear regression. Due to the Faculty of Mechanical Engineering uncertainty associated with the moment of claim occurrence and claim amount if is necessary to secure enough auch to cover the risk Are Dragan LI. Milanovic liability matching requires the prediction of future premium movement pre Bisurance lines which represents the basic concept of development and University of Belgrade operation of insurance companies, Faculty al Mechanical Engineering Erywords: linear regression nonlinear mindeds, prediction, number of policies premimen 1. INTRODUCTION significant data, where as simple methods include only the most important and basic trends in data, thus Definition of business objectives through relation predicting the future more accurately. between the value of income, capenae and pain is one of It is possible to determine the dependence between the basic goals when predicting the cost-effectiveness of variables by their graphic illustration. Le. by drawing the business operations in a specific time frame. Analysis of scatter diagram bussed on the empirical data. Scatter business predictions aims at determining potentially diagram represents the set of values of two observed critical periods that call for additional funds in order to numerical variables, Lc. It shows the movement of provide the continuing operation of a company, function when its argument takes all the values from the This paper deals with predictions and profit domain of definition. Based on the appearance of initial panagement on the example of an insurance company. It values in scatter diagram, the character and intensity of predicts financial results of an insurance company while interdependence of variables may be determined (5.7.12) Celeribing the possible movements of premium amount, The dependence of experimental points cannot when applying linear regression and noelincar models. always be presented with linear model. As a result, this Monitoring and predicting of policy sales flow (premium paper uses different forms of functions to show the flow) is very important in order to provide enough funds to development of regression model. Linear model and cover the risk and expenses and to pencraic profit. exponential, polynomial and logarithmic nonlinear Prodicting the financial results provides the basic concept models are used to predict the number of sold policies of development and characteristics of insurance company's as well as the collected premium amount. Applied operations, It stabilives the company operations, provides curvilinear model can be convened into a model of growth, development and improvement of insurance linear regression by transformation process. Linear market and provides full protection of Insureds' interests. regression is the simplest regression model and it turned 2. HEGRESION MODELS out to be very significant In modelling of a wide variety of phroncena, primarily in marketing and economic research. In addition, it is characterized by a prompt Regression analysis of a phenomenon aleus at defining projection and low complexity level the regularity of a phenomenon developaical. Based on For linear model, sample regression curve has the that regularity, regression analysis enables prediction of form y was, + b, provided that for each pair of sample the future progress of a phenomenon. Regression model data (1. y) i - 1. n, implies that y, max, + b. The goal implies the class of stochastic models represented by an equation where a dependent variable is expressed as a of linear regression analysis is to evaluate coefficients aandh, the describe linear dependence in the whale linear or nonlinear functkm of Independent variables Much research has shown that the complexity of method population, based on the initial date (x. y,] i = 1. n. Parameters of linear trend, a and b, are the variables does not have to be in correlation with data accuracy ( 1- 5,8.13]. Complex methods might get too chase to less calculated for each specific cane and in this paper, they are evaluated by the least squares method:Parameter b represents the initial trend value, while possibility to predict the number of effected policies and parameter a, being the slope of a line. represents the premium amount of the Insurance lines with the biggest constant variable of incline or decline of trend from one share in one of the insurance companies operating in the period to the other. In the mentioned example, those Republic of Serbia (01 - Accident Insurance, 02 - parameters are calculated separately for each line of Voluntary Health Insurance, 03 - Motor Insurance, 07 - insurance, number of sold policies and amount of Goods in Transit Insurance, 08 - Property Insurance collected premium. against Fire and Allied perils, 09 - Other Property Applied exponential model is y m ce ". After app- Insurance Lines, 10 - Motor Vehicle Liability lying the logarithm, the linear model In y ad a + Inc Insurance, 13 - General Liability Insurance, 14 - Creda is obtained, Analysis of the transformed model is the Insurance, 18 - Travel Assistance Insurance, 20 -Life same as the analysis of the linear model. Though, it Insurance and 22 - Supplementary Insurance along with should be pointed out that when interpreting the results, Life Insurance (according to the codes of the National it is important to pay attention to what variables are the Bank of Serbia)). transformed ones. In the mentioned model. values of In order to determine the future flow of collected parameters that are estimated with empincal values, are premium, it is necessary to graphically show the obtained in logarithmic values, since the transformation movement of the number of effected Insurances and of dependent variable was performed. After applying related premiums in the past. Scatter diagrams provide antilogarithm, the value of variable of initial model is clearer picture of phenomenon movement and indicate obtained. the possible type of mathematical model that would The general form of logarithmic regression used for describe the movement of observed phenomenon value prediction is y = p + q In x which is by transformation through the timeframe. In total, 24 scatter diagrams for of independent variable (t = In x)) brought down to 12 insurance lines are created. Two graphs comespond linear form. Analysis of the transformed part is the same to each insurance line. One graph shows the functions of in the analysis of linear model. Nonetheless, special experimental data on number of effected policies per attention should be paid to transformation of policy year, while the other shows the premium amount independent variable. per effected insurance policies. When choosing the curve that beat fits the points in The entire research will be shown through the diagram, the starting point may be the polynomial example of only one insurance line - General Liability regression model. General form of polynomial Insurance (insurance line I1 13). The Figurela) shows regression is y a b, + bat ... + ba". Polynomial the schiller diagram of dependency of number of effected coefficients are the parameters of regression model that policies per policy year, while Figurelb) shows the hould be evaluated. Evaluation is performed with diagram of dependency of number of effected policies In empirical pairs (x., y.). Prediction of parameters bi, i'm (abscissa) and the amount of collected premium in In is determined by least squares method, just like in thousands of Diners (ordinate) of the same policies case of simple linear regression model, having in mind issued in the period from 2008 to 2013 for IL 13. that number of equation should equal to the number of unknown parameters. Second order polynomial regression is used in this paper. Coefficient of determination R . being the relative measure of regression line adjustment to empirical data. indicates the representative quality of a model. It is the relation between the sum of squares deviation explained by regression model and sum of squares of total deviations. Coefficient of determination value in any repression is O S R' S I, It is better if the coefficient is closer to one, which means that the value of residual Figure taj Scatter diagram for IL 13; Number of of ollies par polley you tum is lower, be, the scaner of value around the regression is low, Theoretically, the limit of model's Figure ibj Scatter diagram forL. 13; Number representative quality is set at 0.9. In practice, it is very policies and premium collected according hard to find the model that properly describes that Scatter diagrams created on the basis of initial data phenomenon, therefore, the limit is reduced to 0.6. Law on mummber and amount of effected policies are straight value of coefficient of determination does not always line and curvilinear. For that reason, four regressions mean that graded regression is incorrect Nonetheless, in have been used (linear, exponential, logarithmnke and practical economy it is always better to consider the polynomial) to conuruct a model with independent significance and the meaning of determination variable (years when the policies were effected) and coollicient and to choose the model accordingly. dependent variable (number of effected policies). The aim of this model is to predict future events, in. theof Serbia insurance lines, polynomial regression was the bear Application of linear regression is emphasized in choice, In terms of the remaining insurance lines that this paper since the remaining nonlinear models used in were considered in this poper, the research showed that the paper might be simply brought down to linear form one insurance line had the greatest determination Predicting the total premium amount per insurance lines coefficient when applying exponential regression, for the following year requires that the number of whereas in case of 5 insurance lines non of the effected policies should be estimated based on the regression models was found to be suitable. It may be Historical data. Linear trend applied to predict the concluded that for $84 of insurance lines that were number of effected insurance policies is expressed by included in the research, one of the regrestions might be function y. I am + b. I= 1. in where independent used to predict the premium amount based on the variable x. represents time, whereas dependent variable previously predicted number of policies. In case of one y. represents the number of effected insurance policies out of five insurance lines where none of the regression for each time unit and nvariable of the basic set (n = 6) models could be applyed topredict the insurance The unit for x is one year, whereas for y, the unit is one premium, copcrimental research showed that three out sold policy. Total premium per insurance lines is of four applied models were acceptable for prediction of predicted as a function, Le. regression line, based on the the number of effected policies. This leads to the number of policies calculated in this manner. conclusion that for particular insurance lines(4 in terms of number of policies, 5 in tens of premium amount) 4. COMPARISON MODEL the applied models are unreliable, Having in mind that applied repressions do not correspond to the movement The comparison between the number and the amount of of empirical data, future research may be directed policies obtained by linear, logarithmic, exponential and toward the formation of analytical functions. Their polynomial regression for 12 insurance lines is movement trend is adjusted to the schedule and performed In experimental research. In total, 48 movement of original data in the best possible manner functions were generated to predict the number of and that can be used to show the future sales policies and the same number of functions was created development for those insurance lines separately. The to predict the amount of collected premium. comparison betweenthe results obtained by appli It is determined that linear function best describes of linear regression and nonlinear functions, showed the the movement of empirical data for prediction of deviation of 50#, This deviation indicates that the use number of policies in case of 3 insurance lines, whereas of linear regression concept is valid when predicting polynomial approximation was the most efficient in business results in particular insurance lines. case of 5 out of 12 insurance lines that were considered. This paper compares the methods through the It should be mentioned that determination coefficients example of General Liability Insurance, Figure? shows of Incer and polynomial regressions have the number of sold policies predicted by the approximately the same values in case of 3 out of 5 aforementioned regressions. The highest determination insurance Fries. Therefore, it might be concluded that coefficient of predicted number of effected policies is for 50% of total number of insurance lines considered in obtained in polynomial regression (R = 0:90), while the the research, linger trend turned out to be the best option lowest is obtained through logarithmic regression (R. = for predicting the number of policies soldin the 0.69 which is in practice considered to be the following year. Regarding 4 insurance lines, this representative model). Linear regression model shown experiment showed that linear regression and nonlinear that 83 of change in dependent variable is explained models were unreliable for predicting the number of by the change in independent variable Based on the data policies sold (maximum determination coefficient was projected in this mannerand used to predict the number R' $031 ). Therefore, In case of 67% of examined of sold policies, it is possible to predict the amount of insurance lines, at kait one of the regression models collected premium, Figured, In case of number of might be valid to predict the number of policies. If there policies, as well as in case of premium amount, the best are many regression models whore treads of movement approximation is obtained by polynomial function of are good at adjusting to the schedule and movement of second degree (R # 0.92), whereas the worst (where initial data (which is the case for ] Insurance lines), the determinating coefficient is 0.81) is obtained by prediction model should be chosen cuefully. While logarithmnic function. Relative measure of linear model, applying various regression types to predict number and where the evaluated regression is adjusted to the value amount of policies, it should be emphasized that the of samples, amounts to 0.86. When the deviations of quality of human prediction might be crucial in method many different bead functions based on the empirical selection, in view of the fact that sometimes. dera (in the stated example, the difference in coefficient quantitative methods do not provide better cutimales determination for polynomial and linear regression for than people. The assessment of a manager is significant, number of policies soldis 0,07, while for the premium provided that it is not the single but one of the amount it is 0,06) are small, the choice of method is prediction methods achieved by caperience and assessment of the human Further research deals with central tendency of factor. In the Figures 2 and ], empirical data are movement and development of premium amount based represented by rhomb, whereas predictions (number of on the predicted number of policies, Analysis showed policies and premium amount1. n is determined by least squares method, just like in case of simple linear regression mixlel, having in mind issued in the period from 2008 to 2013 for IL 13. that number of equation should equal to the number of unknown parameters. Second jonder polynomial regression is used in this paper. Coefficient of determination R , being the relative measure of regression line adjustment to empirical data, indicates the representative quality of a model. It is the relation between the sam of squares deviation explained by regression model and sum of squares of total deviations. Coefficient of determination value in any Fig 181 regression is D.SR $ 1. It is bener if the coefficient is closer to one, which means that the value of residual Figure 18) Scatter diagram for IL 13: Number of effector policion par policy you sum is lower, be. the scatter of value around the regression is low, Theoretically, the limit of model's Figure 1bj Scatter diagram forll 13; Number of affected representative quality is set at 0.9. In practice, it is very policies and premium coliccle hard to find the model that properly describes that Scatter diagrams created on the basis of initial data phenomenon, therefore, the limit is reduced to 0,6, Low on number and amount of effected policies are straight value of coefficient of determination does not always line and curvilinear. For that reason, four regressions mean that graded regression is incorrect. Nonetheless, in have been used (linear, exponential, logarithmic and practical economy it is always better to consider the polynomial) to construct a model with independent significance and the meaning of determination variable (years when the policies were effected) and coefficient and to choose the model accordingly. dependent variable (number of effected policies). The air of this model is to predict future events, in. the 3. EXPERIMENTAL RESEARCH direction of policy sales for the following year. The second model that was developed is used to predict the Premium is the biggest source of unemployed financial total premium mount based on the estimated mum wacts of insurance companies, It makes a significant effected policies. Models are created for the above source of investments and ensures financial stability of a mentioned insurance lines according to the data on total country [9,10.11]. Therefore, this paper explores the number of effected policies and total premium in the FME Transactions VOL. 45, No 1, 2017 - 199 period from 2008 to 2013 in a company that deals with that premium torod movement comciponded best to life and nonlife insurance in the teriory of the Republic linear regression in case of ], wherean, in case of 4 of Serbia insurance lines, polynomial regression was the best Application of linear regression is emphasized in choice, In terms of the remaining imurance lines that this paper since the remaining nonlinear models used in were considered in this paper, the research showed that the paper might be simply brought down to linear form. one insurance line had the greatest determination Predicting the total premium amount por insurance lines coefficient when applying exponential region, for the following year requires that the number of whereas in case of i insurance lines non of the effected policies, should be estimated based on the ispiction models was found to be suitable, It may be historical data Lincar inend applied to predict the concluded that for 284 of invurance lines that were number of effected inumance policies in capeeased by included in the research, one of the regressions might be function y. = an + b, I'm I, a where independent used to predict the premium amount based on the variable s. represents time, whereas dependent variable previously predicted number of policies In case of one Yo represents the number of effected insurance policies out of five insurance lines where none of the reprice for each time unit and nyarishle of the basic set (n = 6) model could be applyed topredict the insurance The unit for a is one year, whereas for y, the unit is one premium, experimental research showed that three out sold polley, Total premium per insurance lines is of four applied models were acceptable for prediction of predicted as a function, be. regression line, based on the the mamber of effected policies. This leads to the number of policies calculated in this manner, J conclusion that for particular insurance lines(4 in termsLinear recreation prediction 1500 1.000 1/010 1.100 1000 102 184 15765 1500 1 500 1 2 1 4 5 5 7 Logarithmic regression prediction Ixponential regression production 1500 7 500 7.000 1 LOO 1 2 3 4 5 6 7 2 14 567 Figure 2. Graphically represented prediction of number of insurance policies affected for IL 13 Line of me provision predimina 3501000 GOODDE 100 DOO 360 010 100 030 1.800 7:400 2:500 3:300 3 800 TWOOF TIASO 1:100 1010 Logontoone regression predicia 400 0OD LOLLOOO 750 000 1:800 7 800 2.600 1:100 1500 1:010 Figure 3. Graphically represented prediction of total premium amount based on the predicted under of cold policies for the following year 5. CONSLUSION when predicting than poolincer methodi. Despite the fact that nolinear functions of premium trend in This paper uses the specific example to demonstrigun insurance more realulically describe the mal movement comparison between the results obtained from the use of when compared in the linear, the degive of deviation in central tendency of moveingot and development in queaction icut a well a the determination particular insurance lines while applying lincar coefficient value indicate that appvimotion by use of regression and nonlinear dependencies Conducted experimental research has shown that application of Insurance coarunies ket in the balloonal invcion linear regression is valid in certain insurance lines, it is in financial spurm of a country, Rak dispenion it is significantly simpler to apply linear regression model important kepma ent of their torment. Accordingly. Ush1. After his election, a politician's funds are transferred to a blind trust to avoid possible conflicts of interest. Accordingly, these tu nds are placed at the full discretion of a thirdparty trustee {an individual or an institution], who manages all investment decisions for these funds. The politician does not know where his money is invested; however! he believes that it is invested entirely in Company A with probability {3.25 and in lCompany B with probability [135. The annual return from an investment in |CompanyA is approximately normally distributed with mean 12% and standard deviation 4% whereas the annual return from an investment in Company Bis approximately normally distributed with mean 8% and standard deviation 2%. Assume that the returns from Companies A and E. are independent