Homework: Probability
Many of these questions ask you to create a figure and discuss. When you create tables, charts, and graphs you need to think about formatting. Axes should be labeled, charts should have titles, and so on. You do not have to format in one specific way, but work on making these neat and easy to read. Be careful when copying and pasting documents into word (or similar programs). You may want to paste as a picture. Make sure I can see the chart/table in the document. Please ask me if you have any questions.
Section A: Binomial Distribution
For this first set of questions, suppose you own a romantic restaurant called "Eat Prey, Love" that specializes in serving fresh game. Your restaurant is very popular, and reservations are required. You have 50 tables available and are trying to decide how many reservations to accept. You know from your historical data that 80% of parties that make reservations show up.
I. You are considering taking anywhere between 55 and 65 reservations. On average, you make $20 in profit for each table (after all costs are considered). If a party shows up and there is no table for them (you are overbooked), you will have to give them a $50 gift certificate (you lose $50). - Make a table that shows the expected number of guests (X) and the expected profit for each possible number of reservations (55, 56, ... ,65). Expected profit needs to take into consideration overbooking (hint: use an IF function in Excel).
Make a line chart that shows expected profit based on the number of reservations taken. - Comment on your findings and discuss what number of reservations you think is ideal.
II. Further analysis based on information in QI.
i. Make a table that shows the probability of three different events occurring for each possible number of reservations (between 55 and 65). The three events are: (a) probability you overbook (more than 50 show up), (b) the probability you are less than 90% full, and (c) the probability exactly 50 parties show up.
ii. Discuss the findings of the table. Based on this information, what number of reservations do you think is ideal now? Does your opinion change if you live in a relatively small area and believe that most of your customers are repeat guests? Explain.
III. You are down to your final decision: offer 60 or 62 reservations.
i. Make a table that shows a probability distribution for X for each number of reservations. This should not include all possible values of X. You should group into categories so that we can easily make the comparison.
ii. Make a chart (column, stacked column, pie charts, etc.) that displays the information in your table.
iii. Discuss the distributions and what your final choice would be between these two options (take 60 or 62 reservations).
Section B: Bayes' Theorem and Probability
For this set of problems, assume you are consulting with a health insurance company called "The Nine Circles of Health". They do swab testing for tobacco use and want to know how to interpret the results. From the CDC we know that 15% of adults smoke (cigarettes) regularly, another 10% smoke a few times a month or less (occasional smokers), and 75% are non-smokers. The test the company uses has the following historic conditional probabilities: 90% positive results among regular smokers, 40% positive results among occasional smokers, 10% positive results among non-smokers. The company would like to charge higher insurance rates to regular smokers.
IV. Describing possible outcomes.
Create a table that shows the probability of the 6 distinct joint outcomes (type of smoker and test outcome). A probability tree might help you organize the information.
i. Based on this, assuming those being tested are a random sample of the population, what percentage of tests should the company expect to come back positive?
V. Assessing test results.
i. Create a table that shows the conditional probabilities of smoker type given a positive test result. In other words: what percentage of those testing positive should the company expect are regular smokers? Occasional smokers? Non-smokers? Add another column that shows the same breakdown for those that test negative.
ii. Discuss these results. Based on this, how confident should the company be in charging different rates based on the test results? Assume the company only cares about regular smokers in terms of prices.
VI. Should we do more testing? The company is considering doing a second round of testing to improve the accuracy of the results. This will be expensive, so they want to know how much the additional information would help. Assume the second round of testing is independent, so the probabilities of testing positive based on type of smoker are the same as in the first round.
i. Create a table that shows the probabilities a person that tests positive twice is a regular smoker, an occasional smoker, or a non-smoker. Then add a column with these probabilities for those that test positive one time, and another column for those that test positive zero times.
ii. Based on the probabilities, do you think it would be worth it to test everyone twice? If they do, how should they handle the results of those that test positive one out of two times? (Section C on Following Page)
Section C: The Normal Distribution and price of Apple stock
For this set of questions, we will use the Excel file: "Homework- Apple Stock Price.xlsx". This includes information on the price of Apple stock from January 2015 through the end of 2020 (6 years). There is one sheet that looks at monthly prices, and one that looks at weekly. In each case, I have calculated the percentage change. Note that this is reported on a 0 to 100 scale. For example, in the "Monthly Return" column, the first observation is 6.14. This means Apple's stock price increased 6.14% during the month of January 2015.
VII. Descriptive Statistics
i. Make a table that displays the summary statistics for percentage changes in Apple stock on a monthly basis and weekly basis (just to be clear - each should be in a separate column).
ii. Comment on the difference in returns based on the time period examined. Do these results match what you would expect?
VIII. Normal Distribution
i. Stock returns are often found to follow the Normal distribution. Assuming this is true, use the Excel functions (and the summary statistics from question 7) to find the probability that Apple stock earns greater than a 1% return in a random week. Then do the same for monthly changes.
ii. Use the data to compute the actual probabilities that the return is greater than 1% in a week over the time period. Then do the same for the monthly data. Briefly explain your process for calculating these probabilities.
iii. Make a table that shows the Normal distribution and actual probabilities by time period. - Compare the findings, how well does the data match with the predictions of the Normal distribution?
IX. Suppose your (insane) rich friend wants to invest $100,000 in Apple stock for either one week or one month exactly.
i. Create a table that shows her the probability she will have each of the following outcomes for each of the time lengths: lose more than 5%, lose between 0 and 5%, gain between 0 and 5%, gain more than 5%. Use the normal distribution to calculate these probabilities.
ii. To help your friend visualize the decision, make a graph that shows the Normal probability density function for monthly and weekly returns (two lines on one graph). Display this for the range of -25% to 25%. (hint use NORM.DIST in Excel with cumulative = 0 option).
iii. Based on the information in the table, which would you recommend to your friend? Would your advice be different if your friend tells you she will give you any profit from the investment, but she will expect you to pay her back if the stock loses value? Discuss.
Format your homework according to the following formatting requirements:
i) The answer should be typed, using Times New Roman font (size 12), double spaced, with one-inch margins on all sides.
ii) The response also includes a cover page containing the title of the homework, the student's name, the course title, and the date. The cover page is not included in the required page length.
iii) Also include a reference page. The Citations and references must follow APA format. The reference page is not included in the required page length.
Download:- Apple-Stock-Price.rar