Individual
Modelling, Probability and the Normal Distribution
Question 1
Sandra Enright of Techtronics Inc., an electronics supply firm, has been examining the times required for stock pickers to fill orders requested from inventory. She has determined that individual order-filling times approximately follow a normal distribution with a mean value of 3.9 minutes and standard deviation of 78 seconds.
a) What is the probability that a randomly selected order will require more than three minutes?
b) What is the probability that a randomly selected order will require less than two minutes?
c) What is the probability that a randomly selected order will require between two and three minutes?
d) Sandra is considering a quality assurance that 95% of orders will be filled within a specified time. What time should she specify?
Question 2
Gerald Black of BlackFly Airline has an exclusive contract to run flights of a four-passenger aircraft to a remote mining center. His contract requires him to fly if there are any passengers wanting to make the trip. His fixed costs per day are $300.00, his fixed costs per flight are $1,300.00, the variable cost per passenger is $20.00, and he charges $800.00 per passenger.
He has tracked the number of passengers who flew with him over the past sixty days. His findings are summarized in the following table:
|
Number of Passengers
|
0
|
1
|
2
|
3
|
4
|
|
Number of Days
|
3
|
10
|
17
|
20
|
10
|
Of course, he does not fly on days with zero passengers. Assume that this sample gives a good approximation to his future demand patterns. Let G be the random variable: profit on a future day.
a) Calculate the Expected Value, E[ G], Variance, σ2[ G] and standard deviation, σ[ G], of his future daily profit. [Hint: You can calculate a profit corresponding to each number of passengers. The probabilities of those profits are then determined by the probabilities of the numbers of passengers.]
b) Comment briefly on the profitability and volatility of Gerald's business.
Question 3
Local development initiatives often use estimates of the daily expenditures of tourists to justify expenses incurred in supporting local events. Some years ago, the City of Kingston hosted a Tall Ships weekend, which cost the City some $850,000. To justify this expense, suppose that the City conducted a survey of 30 out-of-town visitors, asking them the grand total of what they spent during their visit to Kingston, and how many days they visited. The data are contained in the file visitor expenditures.xlsx.
a. What is a 99% confidence interval for the average daily expenditure by visitors to Kingston, based on these data? Interpret the meaning of your interval, in simple terms understandable by the Mayor.
b. The Mayor of Kingston at the time, Ms. Turner, had stated that "the average visitor to Kingston spends $200 per day in the local economy". Set up Ms. Turner's comment as a hypothesis test, and use the data to establish whether her statement can be refuted, or not.
c. Some visitors to Kingston are Canadian, and some from other parts of the world. A sample of 200 visitors on this weekend revealed that 120 were Canadian, and 80 from elsewhere. What is a 90% confidence interval for the proportion of visitors who are Canadian on a weekend such as this?
d. The Mayor had also stated that "more than 45% of the visitors to Kingston are Canadians". Set up this statement by the Mayor in a hypothesis test framework, and use the data to determine if her statement can be refuted, or not.
Question 4
Fast Computers Inc. supplies made-to-order personal computers through direct (telephone and online) sales channels. A key competitive feature of its business is the delivery time - the time lapse between receipt of an order and final delivery to the customer. By tracking thousands of previous orders, the company has found that delivery times to the large Toronto market are well-approximated by a normal distribution with a mean value of 4.5 days and standard deviation of 3.14 days. In answering the following questions, you may assume that these parameters are known exactly.
a) What is the probability that a randomly selected order will require more than six days?
b) What is the probability that a randomly selected order will require between three and six days?
c) The company is considering a $50 cash-back guarantee on any orders that have delivery times longer than a specified maximum but are not sure what maximum time to guarantee. They would be willing to pay out the guarantee on no more than 3% of their deliveries. What time limit in whole days should they set for their guarantee?
d) What is the probability that the mean time for a sample of fifteen randomly selected orders will be more than six days?
Question 5 - Reflection
Now that you have a sense for a few specific applications for the material presented is session 1 and session 2 of the course, reflect on the actual or potential use of these techniques in your business. Provide specific examples and comment on your sense as to whether your work place is taking advantage of the opportunities available to collect and analyse statistical data to help inform business decisions.