Two records are to be randomly selected one after the other

Question

1. Two records are to be randomly selected one after the other from an accounts receivable portfolio. The selected records are then examined to see whether they contain any error.
   1. List the experimental outcomes associated with this sampling process.
   2. If the random variable of interest is the number of records that contain an error, what values will this random variable assume for each of the experiment outcomes?
2. Incoming calls to a call centre can have a waiting time of two, three, four, or five minutes before they are answered. The different waiting times occur at about the same frequency.
   1. Develop a probability function for the waiting time of an incoming call.
   2. Draw a graph of the probability distribution.
   3. Show that the probability function you developed in Part 2a satisfies the condition required for a discrete probability function.
3. The following is probability function used to describe the numbers of times (x) a student receives an A in the two project assignments of a statistics course:
   f(x) = (3-x)/6 for x=0, 1, 2
   1. Is this probability function valid? Please explain.
   2. What is the probability that a student receives exactly one A in the two project assignments?
   3. What is the probability that a student receives at least one A in the two project assignments?
   4. What is the expected value of this probability distribution?
4. A market research study on buyers of a condo project recorded the number of times a buyer visited the project web site before the purchase decision was made:
   

   Number of times web site was visited	Number of buyers
   0	245
   1	1245
   2	2361
   3	1893
   4	876

   Let x be the random variable representing the number of times the project web site was visited before the purchase decision was made.
   
   1. Show the probability distribution function of this random variable
   2. What is the probability that a buyer visited the web site exactly two times before the purchase decision was made?
   3. What is the probability that a buyer visited the project web site three or more times before the purchase decision was made?
   4. What is the expected value of the number of times the project web site was visited before the purchase decision was made? What is the interpretation of the expected value?
   5. What is the variance and standard deviation for the number of times the project web site was visited before the purchase decision was made? Please be accurate within 2 decimal places for these two calculations.
5. ABC Inc. is a large Canadian company with thousands of employees across the country. Its recent study on employee engagement showed that 75% of the employees participated in the study agreed with the statement that their managers provide them with the necessary support on a consistent basis. The company believes that employees included in the study are representative of the entire employee base. Suppose that 12 employees are selected randomly from the employee base.
   1. Is the selection of 12 employees a binomial experiment? Please explain.
   2. What is the probability that 10 of the 12 employees selected agreed with the statement that their managers provide them with the necessary support on a consistent basis?
   3. What is the probability that at least 10 of the 12 employees selected agreed with the statement that their managers provide them with the necessary support on a consistent basis?
   4. What is the probability that at most 4 of the 12 employees selected agreed with the statement that their managers provide them with the necessary support on a consistent basis?
6. Historically, 45 percent of donors to a charitable foundation were male. In a recent study, a sample of 10 donors is to be randomly selected from its donor list which contains the record of 3500 donors.
   1. What is the probability that none of the donor selected is male?
   2. What is the probability more than 2 but at most 6 of the 10 donors are male?
   3. What is the probability that less than 4 of the 10 donors are female?
7. Incoming calls to a call centre is at a rate of one every 2 minutes. Use the Poisson probability distribution to answer the following questions:
   1. What is the expected number of calls in one hour?
   2. What is the probability of three incoming calls in six minutes?
   3. What is the probability of at least one call in a six-minute period?
8. At ABC Bank, customers arrive at a teller line at a rate of 10 per five-minute period. Assuming that customers arrive randomly and independently, use the Poisson probability distribution to answer the following questions:
   1. What is the probability that no customers arrive in a five-minute period?
   2. What is the probability that 3 or few customers arrive in a five-minute period?
   3. What is the probability that no customers arrive in a one-minute period?
   4. What is the probability that at least 1 customer arrive in a one-minute period?