Bob's Grocery Store is in need of some help. The store manager, Bob, has four troublesome worries and he has hired you for your probabilistic prowess. Bob has a reasonable understanding of basic probability but needs things explained in a rather thorough manner. Clear, well thought out, and nicely summarized results make Bob happiest. He doesn't necessarily need you to make decisions for him, so if you have more than one possible solution to any of his problems, include them all, and Bob will make the final decision based on his feelings and your presentation of the various solutions.

The first problem involves trying to save a bunch of money from a seafood distributer. Loey's Lobsters has informed Bob that if he can place a fixed bi-weekly order for Lobsters, the price will be significantly reduced. Thus, Bob needs to figure out how many Lobsters, say L, he will sell each two week period, for a total of 16 weeks. During this 16 week period, Loey will deliver exactly L lobsters every two weeks, and Bob will save a bundle (assuming L is chosen appropriately). Bob isn't quite sure what L should be however. Lobsters don't keep too well (Bob sells the live ones), but Bob makes a good profit on them, so he can't afford to run out very often either. Ideally, Bob would like to be able to choose L so that he sells at least 80% of the lobsters each two week period, and runs out in a given 2 week period with probability less than 10%. A two-week old lobster must be disposed of. You should try to estimate the total expected number of lobsters that Bob will have to dispose of for the entire 16 week fixed-order period. Bob, being the fanatic he is for detail, has recorded the number of lobsters sold each week, for the last 49 weeks. This data is given below. By considering the data, give Bob a value (or two) for his bi-weekly order amount L. Click here to download an Excel file with the data.
 
 

The second problem has to do with an idea that Bob has to attract new customers to his store. He figures that if he consistently has a low price on a common item, like Pepsi products, shoppers who are not devoted to any particular store might just pick Bob's store to save a few cents on pop. Pop prices fluctuate quite a bit, and Bob would again like a fixed price that is close to the 5th percentile of pop prices in his city. That is, he would like to fix a price P, which in the long run would be as cheap or cheaper than 95% of the competitors prices on Pepsi (2-liter bottles anyway). Bob realizes that certain chains of grocer's regularly put Pepsi on sale. He too can occasionally lower his standard price for such sales. Bob again has collected a few months worth of Pepsi prices at local stores. Some of these are obviously sale prices, and it would probably be best to ignore any excessively low prices when trying to calculate a value for P. Here are the prices (in dollars) that Bob has collected over the past few months.
 
 

Next, there are a few rare items that Bob sells in his store, like Zud (a rarely used cleanser), which are not stocked by Bob's warehouse. Usually Bob can order an item one day, and it will arrive on the next warehouse delivery (which occurs 3 times each week). However, for these non--warehouse items, it can take several weeks before the item arrives. In the case of Zud, it takes about 20 days (from when Bob places the order) for the order to arrive. A case of Zud has 24 cans, and Bob has enough room on his shelf to hold 32 cans of Zud. Since Zud sells so infrequently, a partial case (that won't all fit on the shelf) that needs to be stored in the backroom until more cans sell, makes Bob look pretty foolish. His employee's would riddle him to death for such a slip--up in ordering. Obviously, Bob could wait until only 8 cans remained on the shelf, but this causes two possible problems. First off (and perhaps the most embarrassing of all grocery store errors imaginable), the 8 cans of Zud could sell before the new case arrives (Zud buyers would boycott Bob's store forever). Secondly, since Bob likes his shelves to look full, if he has less than 4 cans of Zud (his Zud display on the shelf is 4 cans wide in the front), there will be a sinful empty spot on his shelf once he has fewer than four cans of Zud remaining. For these reasons, Bob would like to know the number Z, so that when only Z cans of Zud are left on his shelf, he can safely order a new case. The number Z should be chosen so there is as little likelihood as possible of the new case arriving before there is room on the shelf; or the case arriving after Bob has 3 or fewer cans of Zud left in his store. Suggest a choice or two for Z, and comment on the likelihood of running out (completely) of Zud, running short (3 cans or less remaining) or needing to store a few extra cans that won't fir on the shelf in the backroom, using these Z values. When Bob last ordered Zud, he electronically tagged all 29 cans that he had on the shelf, and recorded the number of days that passed before each can was sold. These time periods are given below.
 
 

Note that can 1 is simply an abbreviation for the first can to sell; can 2 was the second can to sell and so on. Also note such things such as can 4 sold 5 days after can 4 and 3 cans sold in the first week. There is a variety of information encoded by the above table.

Finally, Bob has sold 512 tickets to his store picnic and needs help figuring out how many hot dogs to order for this event. Picnickers can eat either hot dogs or hamburgers (or both) and past experience indicates that for every 5 picnic go-ers, Bob needs an average of 3 hot dogs. The number of hot dogs eaten varies a fair bit however and Bob would like to order Z hot dogs, so that the probability of running out this year is about 2% (and the likelihood of having many extra hot dogs is small).

To help Bob out, give him your best estimates for values of L, P, Z and H. When choosing these values, pay special attention to the type of distribution (normal, Poisson etc.) that most likely controls each value and the methods you use to estimate the parameters of each distribution. The reasons for your choices of the type of each distribution and the parameter values for that distribution should be the focus of your report to Bob. You may need to make histogram plots of some of the data to determine the type or parameters. Some of the data is not presented in its most useful form. You should feel free to manipulate the data however you see fit, including the possibility of ignoring any data which seems to not fit the pattern. Keep in mind that you have a large amount of data to test your conclusions about the distribution types and parameter values. For example, if you decide that 2-liter Pepsi prices (this is actually a silly-made up choice) are binomial with n = 300 and p = 0.35, then you can calculate the probability that such a binomial random variable takes on values between 95 and 110, and compare that to the proportion of actual Pepsi prices which lie in this range. Don't expect all your test calculations to be too accurate, but they should give reasonable results when compared to the actual data. Don't be afraid to use discrete random variables to approximate continuous quantities, or continuous random variables to approximate discrete data; both practices are common and sometimes work well. While you can answer each of Bob's questions using simple averages or proportions, this will receive a rather mediocre score; it amounts to assuming that all quantities are uniformly distributed, which is not the case here. This is by far the most time demanding assignment of the semester. I would strongly suggest you start working on it soon.

The class will be divided into four groups to complete this assignment. The last day or two of class will consist of group presentations on your findings. I will randomly select a letter (L, P, Z or H) and then randomly select a group and a random member of that group to explain their choice for the selected letter. The entire group will receive the grade assigned to the randomly selected person's presentation. You should take very seriously the charge of ensuring that each member of your team fully understands the work you do to select a value for each letter.