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then the data must be interpreted and then entered by the data-entry staff This process generally results in the least accurate data of all, for there are now two people entering information (the employee and then the data-entry person), creating two opportunities to make a mistake In short, the best way to avoid charging time to the wrong job is to have an interactive data-entry system Another problem is that vastly inaccurate amounts of hours will sometimes be charged to a job, usually through an inaccurate recording of numbers For example, an eight-hour shift might be entered incorrectly as 88 hours To avoid such obvious mistakes, the timekeeping system can be altered to automatically reject any hours that clearly exceed normally boundaries, such as the number of hours in a shift or day A more sophisticated approach is for the timekeeping system to automatically accumulate the number of hours already charged during the current shift by an employee This system yields an increasingly small number of hours that can still be worked through the remainder of the shift any excess can either be rejected or require an override by a supervisor (indicating the presence of overtime being worked) This approach is not possible if employees record their time on paper, since the information is entered after the fact and any correction of an incorrect number will be a guess, sometimes inaccurate, by the data-entry person Another possibility is that an employee charges an incorrect employee code to a job resulting in the correct number of hours being charged to the job, but at the labor rate for the employee whose number was used, rather than the rate of the person actually doing the work To avoid this issue, the timekeeping system should automatically access a list of valid employee numbers to, at least, ensure that any employee code entered corresponds to a currently employed person Though this is a weak control point, it at least ensures that hours charged to a job will be multiplied by the hourly labor rate of someone, rather than zero A much better control is to require employees to use a bar-coded or magnetically encoded employee number that they carry with them on a card They are forced to enter the same employee code every time A lesser control is to post a list of bar-coded or magnetically encoded employee numbers next to each data-entry station this is a less accurate approach, since an employee can still scan someone else s code into the terminal If a paper-based system is used, an employee normally writes his or her employee number at the top of a time report, which is then entered by a data-entry person into the computer at a later date The problem with this approach is that the data-entry person may enter the employee number incorrectly, which will charge all of the data on the entire time report to the wrong employee number Once again, an interactive timekeeping system is crucial for the correct entry of information Yet another problem is that the cost per hour that is used by the timekeeping system may not be the same one used in the payroll system This problem arises when there is no direct interface between the timekeeping and payroll systems Costs per hour are only occasionally (and manually) transferred from the payroll system to the timekeeping system This results in costs per hour on timekeeping reports that are generally too low (on the grounds that employees generally receive pay increases, rather than decreases, so that any lags in data entry will result in costs per hour that are too low) One way to fix this problem is to create an auto-.

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Combinations Table 25 n 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 Probability 00833333 0136364 0159091 016317 0157343 0146853 0134615 0122172 0110294 00993292 00893963 00804954 00725678 00655294 00592885 00537549

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Now let us compute a table of values of n and these probabilities, letting m = 8 for various values of n This is shown in Table 25 A graph is helpful (Figure 23) We see that the maximum probability occurs when n = 13, so we have found the maximum likelihood estimator for n The mathematical solution of this problem would be a stretch for most students in this course The computer is of great value here in carrying out a fairly simple idea It should be added here that this is not the optimal solution for the German tank problem It should be clear that the maximum value in the sample carries more information about n than does the median We will return to this problem in 12

I I Cont i n u e ( whoop s ! )