Homework 4

1. Jessica Williams, manager of Kitchen Appliances for the Midtown Department Store, feels that her inventory levels of stoves have been running higher than necessary. Before revisiting the inventory policy for stoves, she records the number sold each day over a period of 25 days, summarized below.

number sold 23 4 5 6

number of days4 7 8 5 1

For example,there were 4 days out of 25 days, in each of which 2 units were sold;there were7days out of 25 days, in each of which3 units were sold;so on and so forth.

a. If we define a random variable X as the number of units sold per day, use these data to estimate the probability distribution and cumulative distribution for X in a tabular format, and calculate the Expectation of X. (hint: if you are given a loaded (i.e., biased) die, how would you estimate the chance of each face turning up?)

b. Currently, Jessica is keeping 50 units in stock, and she will place another order of 45 units once the inventory level drops to 5 units. How frequent does Jessica place an order each month (25 work days)? (Hint: use the notion of expectation.)

c. The profit for each stove sold is $150. For each unit sold, there is a 5% chance that the unit will be returned by the customer for full refund. The returned unit will then be sold to a discount store at a lower profit of $50. What is the expected daily profit from stove sales? (Hint: define another variable Y as the actual profit for each unit sold taking into account the returns.)

2. Identify probability distribution for each of the following cases. Are they Binomial, Poisson, Normal, or Uniform distribution. Why?

a) A professor receives, on average, 24.7 e-mails from students the day before the midterm exam. To compute the probability of receiving at least 10 e-mails on such a day, he will use what type of probability distribution?

b)What type of probability distribution will the consulting firm most likely employ to analyze the insurance claims in the following problem?An insurance company has called a consulting firm to determine if the company has an unusually high number of false insurance claims. It is known that the industry proportion for false claims is 3%. The consulting firm has decided to randomly and independently sample 100 of the company's insurance claims. They believe the number of these 100 that are false will yield the information the company desires.

c) What type of probability distribution will most likely be used to analyze the number of blue chocolate chips per bag in the following problem?The quality control manager of a candy plant is inspecting a batch of chocolate chip bags. When the production process is in control, the average number of blue chocolate chips per bag is 6.0. The manager is interested in analyzing the probability that any particular bag being inspected has fewer than 5.0 blue chocolate chips.

d)A company has 125 personal computers. The probability that any one of them will require repair on a given day is 0.025. To find the probability that exactly 20 of the computers will require repair on a given day, one will use what type of probability distribution?

e)A multiple-choice test has 30 questions. There are 4 choices for each question. A student who has not studied for the test decides to answer all questions randomly. What type of probability distribution can be used to figure out his chance of getting at least 20 questions right?