Probability （graduate level）

Please show and explain your work, including which functions you used in Excel – do not just write the answers.

1. MBB Problem 7.2 (modified)

The average (mean) amount of time that a manager at the Cimaron Valley Department of Human Services spends in the annual performance review with an employee is 27.5 minutes, with a standard deviation of 2.5 minutes (normal distribution).

(a) What percentage of the annual performance reviews in the department take between 25.0 and 30.0 minutes?

(b) Between 22.5 and 32.5 minutes?

(d) You can add two normal distributions together and the new distribution is also a normal distribution with a mean equal to the sum of the two means and a standard deviation equal to the square root of the sum of the two square roots (assuming that the two distributions are independent of each other). So, if the manager did two performance reviews in a row, the two reviews combined would last a mean of 27.5 + 27.5 = 55.0 minutes with a standard deviation of = 3.54 minutes, and if the manager did three reviews in a row, the three reviews combined would last a mean of 27.5 + 27.5 + 27.5 = 82.5 minutes with a standard deviation of = 4.33 minutes. If the manager schedules four performance reviews in a row every day starting at 10:00, what percentage of days would she expect to finish before 12:00.

2. MBB Problem 7.4 (modified)

Refer to Problem 7.3 (which we covered in lecture). The head of the Mariposa County Accounting Department wants to establish a standard regarding the length of time that employees can expect to wait to receive reimbursement for professional expenses.

(a) She wants to publish a standard that states the maximum number of days it takes the department to process 95% of the claims filed. Help her find that standard.

(b) Instead of the maximum number of days, she considers publishing a standard that states the range of days it takes the department to process 95% of the claims filed. Help her find that standard.

3. MBB Problem 7.10 (modified)

The director of development at the Richman Children’s Science Museum has recently collected data on donations for the past several years. She finds that the data are normally distributed with a mean of $51 and a standard deviation of $14. The director’s ideal point for a minimum donation is $60.

(a) What percentage of individual donations are less than $60?

(b) What percentage of individual donations are $60 or more?

(c) The director’s long-term goal for an average individual donation is at least $80. Based on the current percentage of donations that meet that goal, does achieving this goal seem reasonable in the immediate future?

4. MBB Problem 8.4 (modified)

In a grand jury case, a bookstore was indicted in Oklahoma County on several counts of selling an obscene book. The grand jury was composed of 22 Baptists and 8 other people. The defendant feels that Baptists are biased against free speech.

(a) What is the probability that 22 or more Baptists are selected on a jury if Oklahoma County is 40% Baptist?

(b) Although the entire population is only 40% Baptist, the population available for jury duty is 60% Baptist. What is the probability that 22 or more Baptists are selected for the jury if the available jury duty population is 60% Baptist?

5. MBB Problem 8.16 (modified)

The Bluefield Regional Employment Service needs to place five individuals in jobs this week to meet its yearly quota. During the past several years, the service’s track record is that every person sent to interview for a job has a .6 probability of getting the job. The placements appear to be independent of each other.

(a) The service decides to send seven individuals for interviews this week. Based on what you know, what is the probability that the service will make its yearly quota this week?

(b) The service would like to be at least 90% sure of meeting its quota. If it sends eleven individuals for interviews this week, can it be 90% sure?

6. MBB Problem 9.10 (modified)

The mean number of contracts that consulting firms receive with the Department of Human Services is .5 for any given year.

(a) Ecosystems Inc., a heavy contributor to political candidates, receives two contracts in one year. What is the probability that this event would occur by chance?

(b) What is the probability that Ecosystems will receive four contracts in two years?

7. MBB Problem 9.16 (modified)

The University of Wisconsin employs 200 teaching assistants (TAs). If one-third of the TAs sign a petition calling for a collective bargaining election, an election will be held.

(a) A survey of 25 TAs indicates that 40% will sign the petition. What is the probability that a sample such as this could have occurred if one-third or fewer of the TAs in the population will sign such a petition?

(b) The university surveys 25 additional TAs, so they now have a survey of 50 TAs, which also indicates that 40% will sign the petition. What is the probability that a sample such as this could have occurred if one-third or fewer of the TAs in the population will sign such a petition?