Page 1 of 13LIR 832:Mid-term Examination: Fall, 2005Each Problem is worth 20 points. Answer five of the six problems. Problemsshould be answered as thoroughly as you are able. Partial credit on problems is onlypossible if I can locate errors in your calculations; neatness and organization of youranswers is essential Remember, answer only five of the six problems, if you choseto answer all, I will select the five with the lowest scores.1. Employee morale is affected by working conditions and amenities at thework site. Although it is possible to over do the amenities, betterconditions – lunchrooms that are clean and have enough space for thelunch rush, sufficient numbers of clean bathrooms, good ventilation andtemperature control – can be shown to improve employees views of workand employee performance.Although it seems silly, bathrooms play a critical role in employee attitudesand the efficiency of a workplace. If there is too little bathroom space(aka, number of toilets), people spend considerable time waiting for atoilet. They are typically not happy about this and work time is lost whilemultiple employees wait. On the other hand, bathrooms are expensiveboth in terms of direct cost, floor space and cleaning time (the only thingworse than waiting for a toilet is waiting for a dirty toilet).We are planning a new work site with about 100 employees. Theplanners for the site have asked about bathroom size and the number oftoilets. Our records indicate that, on average, 3 toilets are in use at anyone time (don’t ask how we know this).Given the trade off between the number of toilets and the costs ofbathroom facilities, we would like to have enough toilets so thatemployees will only have to wait for a stall 10% of the time.A. What type of probability distribution should we use to calculate theprobabilities for the number of toilets in use? Explain why this is thecorrect distribution.Poisson distribution, which gives us the probability of a number of events occurring in a fixed time period.Page 2 of 13 B. Calculate the probabilities that 1, 2 3, 4, 5 and 6 toilets are in usemean of x 3exp(-3) 0.049787068367864Number of Calls(aka x)mean(x)^x x! P(x) CumulativeProbability0 1 1 0.04979 0.049791 3 1 0.14936 0.199152 9 2 0.22404 0.423193 27 6 0.22404 0.647234 81 24 0.16803 0.815265 243 120 0.10082 0.916086 729 720 0.05041 0.96649C. How many toilets do we need to be sure that employees will have to wait for atoilet only 20% of the time? How many do we need to be sure that employeeshave to wait only 10% of the time.We need to use the cumulative probabilities for this. The likelihood that 4 orfewer toilets will be in use at any time is 86%, so that meets our 81.5% goal(20% of the time they have to wait). The likelihood that 5 or fewer stalls willbe in use if 91.6%, so with 6 toilets we should hit our 10% goal.Page 3 of 132. A global organization with its headquarters in Germany absorbed an American firmseveral years ago. There is concern among those working for the American subsidiarythat they are less likely to be promoted into management (even of the subsidiary) thanthose working for the parent company. You have been asked to analyze data onpromotions to determine if those in the American subsidiary are, statistically speaking,disadvantaged with respect to promotion into management.You know that 40% of employment in the global organization is in the Americansubsidiary. Considering the last 15 promotions, you find that 3 went to Americans andthe remainder went to Germans. What is the likelihood that this outcome was theproduct of a process that was blind to nationality?A. What type of distribution should you use to calculate this probability? Why?BinomialB. What is the probability that we would have gotten the observed outcome (3 out of 15Americans) if the selection was blind to nationality. What do you conclude from thistest? Why?pof()!!* !*. *. .3151531246 634%312==Page 4 of 133. Extended overtime affects worker efficiency. Since Captain Jones research on theproductivity differences on 8 and 12 hours shifts at Carnegie Steel in the 1880s, hundredsof studies have found that, absent particular circumstances, working beyond 40 per weekresults in lower productivity per hour or per worker.We have data on hours of work and the value of output in the last hour of work formechanical contractor (aka, plumbers and Pipefitters)across firms that work different schedules. Firm AverageHours ofWorkValue of Hourly Outputper Employee for theLast Hour of Work135 24240 19343 10447 7540 25A. Calculate the covariance between average weekly hours of work and thevalue of output. Is the estimate consistent with the view that workinglonger hours causes marginal (last hours) output to decline? Explain.Page 5 of 13Page 6 of 13B. Calculate the correlation between the average hours and the value ofhourly output. How strong is the relationship between hours and value ofoutput? What does correlation tell you about this relationship that youcould not learn from the covariance measure?Covariance and Correlation When You have 'raw' data:X Y (X - XBar) (X -Xbar)^2 (Y -Ybar ) (Y -YBAR)^2(x-xbar)*(y-ybar)35 24 -6 36 7 49 -4240 19 -1 1 2 4 -243 10 2 4 -7 49 -1447 7 6 36 -10 100 -6040 25 -1 1 8 64 -8sum = 205 85 sum = 78 sum = 266 -126avg= 41 17 var(x)= 19.5 var(y)= 66.5 COV(X,Y)= xy-31.5std dev(x) 4.41588 std dev(y) 8.15475Corr(X,Y)= -0.87475Page 7 of 134. Diversity is important to our firm and hiring a workforce that “looks like Michigan” (wewon’t even begin to fully explore the implications of the slogan), is the byline of ourrecruiting effort. One dimension of diversity is a more balanced representation of thecolleges in Michigan. Because the firm is located in Lansing, we have traditionallyfavored hiring graduates of the MSU and have tended to stay away from graduates of theUofM. No other school in Michigan is worth considering, and so we don’t. The schoolseach graduate similar numbers of undergraduates and graduate students, so thecomposition of the Michigan workforce with college degrees is evenly split between MSUand MU graduates. The mean for the percentage of college graduates who are fromMSU for the Michigan workforce is .5 with standard deviation of .25.Despite our stated goal of encouraging diversity in college background, the old school tieis strong among our current workforce and we are concerned that they may be continuingto favor MSU grads. We look at