Statistics and Quantitative Methods for Managers

1. An airline has four ticket counter positions at a particular airport. The airline has set a new goal of limiting the waiting times for its customers at the counters to not more than 5 minutes. As part of its effort to reduce the waiting time, the airline has introduced the ‘snake system’. Under this system, all customers enter a single waiting line that winds back and forth in front of the counter. A customer who reaches the front of the line proceeds to the first free position. Each weekday for six weeks, the airline customer-relations manager charts the waiting time in minutes for the first customer entering after 4:00 PM. The resulting data show a sample mean waiting time of 4.2 minutes with estimated population standard deviation of 1.1 minutes. Using the given information please calculate and interpret the 92% confidence interval for the mean waiting time of customers of the airline. Please show the necessary steps. Based on your results, is it reasonable to conclude that the airline is achieving its stated goal? Please justify your answer.                                    [2 points]
2. Winta Temp Services Agency finds it difficult to retain its employees because most of them are looking for full-time positions and will leave the Winta Agency when a good opportunity comes along. For 100 most recent Winta employees who terminated, the average time of employment was 7.5 months. Assume that this is a random sample of Winta employees. If the population standard deviation for the duration of employment of Winta  employees is estimated to be 8.5 months, do you think it will be reasonable to conclude that an employee who stayed with the company for 12 months stayed “unusually long?” To answer this question accurately, please use one of the conventional confidence levels with the largest possible margin of error. Please show the necessary steps and interpret your result.                                                                          [2 points]
3. Baratieri Pizza Restaurant is facing a stiff competition from the new competing pizza restaurant guaranteeing pizza deliveries within 30 minutes or the pizza is given free. To answer this challenge, Baratieri  wants to offer a 28-minute delivery guarantee. After a careful cost analysis, Baratieri  has determined that such a guarantee would require an average delivery time of less than 26 minutes. She thought that this would limit the percentage of ‘free pizzas’ under the guarantee to less than 10.5% of all the deliveries, which she had figured to be the break-even point for such a promotion. To find out if the restaurant can meet these requirements, Baratieri  collected data on the total delivery times and ‘free pizzas’ for a random sample of 81 orders within a month. The total delivery time includes the preparation-time, the wait-time, and the travel-time for the drivers. The sample data show an average total delivery time of 24.9 minutes with estimated population standard deviation of 1.2 minutes. The data also show that 4 out of the 81 orders resulted in ‘free pizza’ deliveries because they took longer than 28 minutes to deliver. Baratieri  is using a non-conventional confidence level of 93% in her estimates. Please answer the following questions using the information given in this problem.
4. Please calculate and interpret the 93% confidence interval for the mean total delivery time of pizzas for the Baratieri Please show the necessary steps.              [2 points]
5. Please also calculate and interpret the 93% confidence interval for the proportion of ‘free pizzas’ expected under Baratieri’s proposed promotion scheme. Please show the necessary steps.    [2 points]
6. Based on the results for the two confidence intervals you calculated above, do you

think Baratieri’s proposed promotion scheme is viable? Please carefully justify your answer.                                                                                               [1 point]

4. A state fish hatchery raises trout for stocking streams and lakes. The size of the fish at release time can be controlled to a fair degree by varying the rate of feeding. The target is a mean of 10 ounces: if the fish are too small, those who catch the fish aren’t too happy, but if the fish are too large, those why buy the feed aren’t too happy. A random sample of 67 fish are weighted at release time. The weights, to the nearest tenth of an ounce, are stored in column 1 of the Excel data file named “Fish Weights.”
5. At the 93% and the 97% confidence intervals, do you think the hatchery is meeting its goal? Please clearly show all the necessary steps and interpret your results.                                                [3 points]
6. What is the probability that the true mean weight of the fish could be less than the upper limit of the 93% confidence interval? Also explain which confidence interval is wider and why. What do you think would happen to the confidence intervals if the sample size were to be increased to 100? Please justify your answers.              [2 points]

5. A mortgage service company processes a large volume of transactions every day. Because many of the transactions involve depositing funds, it is important for the company to complete processing and not leave the unprocessed items for the following day. The operations manager has a target that processing will be complete in 95% of all workdays. One of her assistant managers has been using an Excel data file where he has created a variable named “AllDone”. For the last 160 days, he has been entering a value of 1 for this variable when complete processing was achieved and 0 when unprocessed items were left for the next day. At the end of the 160^th day there were 14 zero entries under the AllDone variable and the rest were all ones. After a preliminary analysis of the data, the operations manager has concluded that “we are running short of our target and we need to step up!” At the 92% confidence level do you agree with the manager’s conclusion? Does your conclusion change at the 97% confidence level? Please show the necessary steps and interpret your results in each case.                                                             [2 points]
6. An investment banking firm was considering two spreadsheet software for possible use in its large data analysis operations. A sample of 28 typical analysis problems were carried out using both software on the firm’s standard personal computer. The time in minutes needed to load the data, program the spreadsheet calculations, carry out the analysis, and print the results was recorded for each problem for each software. The data for software A are stored in column 1 of the Excel data file named “Banking Spreadsheet.” The corresponding data for software B are stored in column 2. Each of the rows corresponds to one of the 28 problems. After constructing the 96% and 93% confidence intervals for the mean of the differences in the minutes taken by the two software, a data analyst for the company concluded that there is no substantive difference between the two software in terms of the speed with which they process the data. In other words, he thought the mean of the differences in the minutes taken by the two software is statistically negligible. As a result he decided to recommend software B because he thought it is more user-friendly than software A. Using the same data, please calculate and interpret the confidence intervals upon which the analyst’s recommendation was based. Based on your results would you agree with the analyst’s conclusion? Please justify your answer.                                           [3 points]
7. Satcon agricultural equipment retailer sells and services several brands of tractors. The data on demand for the last 400 weeks and lead time for the last 50 orders have resulted in the following frequency distributions for a particular model of tractors.

The company is trying to determine a cost-minimizing strategy for its tractor inventory management. They are currently considering the following two strategies:

3. Order quantity of 20 with a reorder point of 3.
4. Order quantity of 15 with a reorder point of 5.

The holding cost is $10.00 per week for each unit that is left in inventory at the end of the week. The stock out cost is estimated to be $400 per stock out and the ordering cost is $500 per order.  Please simulate 35 weeks operation for Tracon assuming there are currently 8 units in inventory. Please report all the tables with your results as part of your submission for this problem.

Note: Use the same random numbers for the two strategies so that differences in the simulation results then solely reflect the differences in the selected values of controllable variables  – not random variation in the scenarios used.

Based on your simulation results please determine the weekly stock out cost, ordering cost, weekly holding cost, and the total inventory cost for each of the two inventory management strategies under consideration. Which strategy will be more cost-effective for the company?                                                                                            [10 Points]