251grass3-08b 6/12/08 (Open this document in 'Page Layout' view!)
Graded Assignment 3
Name:
1. (Source 03) Find for the following distributions (Use tables in d, e, f, h and j.):
a. Continuous Uniform with (Make a diagram!).
b. Continuous Uniform with (Make a diagram!).
c. Continuous Uniform with (Make a diagram!).
d. Binomial Distribution with .
e. Binomial Distribution with .
f. Binomial distribution with (Approximate Solution)
g. Geometric Distribution with
h. Poisson Distribution with parameter of 9.
i. Show how you would do this for a Hypergeometric Distribution with
Remember .
j. Hypergeometric Distribution with (Approximate Solution)
k. For a Hypergeometric Distribution with , find . (Extra credit if you actually evaluate this.)
l. (Extra credit) Findfor an exponential distribution with .
2. Find the Mean and Standard deviation for the following distributions.
a. Continuous Uniform Distribution with .
b. Binomial Distribution with .
c. Geometric Distribution with
d. Poisson Distribution with parameter of 9.
e. Hypergeometric Distribution with .
f. Compare means and standard deviations for a Binomial distribution with and a Hypergeometric Distribution with .
g. (Extra credit) Exponential distribution with
3. Identify the distribution and do the following problems. Use tables where possible.
a. This problem uses all four discrete distributions.
(i) An IRS auditor is about to audit 6 returns from a group of returns for improper deductions.
Assume that these returns come from a group of 80 returns in which 25% have improper deductions, what is theprobability that exactly one of the 6 has improper deductions?
(ii) What is the probability that at least one of the six has improper deductions?
(iii) If two or more of the 6 returns have improper deductions, the entire group of 80 will be audited. What is the chancethat that happens?
(iv) Assume that the 6 returns come from a large group of returns (more than 120) that is 25% improper, and if two ormore are improper, the whole group will be audited, what is the probability that the whole group will be audited? (Answersthat use a specific population size will not be counted.)
(v) Assume that a large number of returns are audited from this group that is 25% improper, what is the chance that the second return is the first one found with improper deductions?
(vi) What is the chance that the first improper return will be among the first ten audited?
(vii) Assume that only 2% of the returns are improper and a sample of 100 taxpayers is taken, what is the chance that at least 3 have improper returns?
b. A fast food hamburger must be cooked a minimum of 90 seconds to be safe. At my lunch spot the time that burgers are on can be represented by a continuous uniform distribution with limits of 80 and 120 seconds. (i) What is the chance that a hamburger will be safe? (ii) What is the chance that, if I order 2 hamburgers both will be safe? (iii) What is the chance thatif I order 3 hamburgers, at least one will be unsafe?
c. The number of students who come to my office has a Poisson distribution with a parameter (mean) of 2.5 a day. (i) What is the chance that no students come in in a day? (ii) In a 5-day week? (iii) What is the chance that more than 5 come in in a day (iv) in a 5-day week?
d. Complaints come in to my website at an average of seven an hour and take an hour to adjust. How many complaint handlers do I need to keep the probability of a complainer having to wait below 1%? (Hint: Look at a Poisson table with a mean of 7. What is the probability thatis above 10?)
e. If 80% of shoppers at a website abandon their shopping carts before quitting and there are currently 20 shoppers on my site, what is the chance that at least one will buy something:? What is the average number that will buy something? What is the most likely number that will buy something?
f. According to Bowerman and O’Connell, in the movie Coma a young nurse finds that 10 of the last 30000 patients of a BostonHospital went into a coma during anesthesia. Upon investigation she finds that nationally 6 out of 100000 patients go into comas after anesthesia. Assume that the national probability is correct and that the Binomial distribution is appropriate, what is the mean number of patients out of 30000 that will go into a coma? In a hypothesis test, a researcher finds the probability of getting a number as large as or larger than what actually happened. So, find the probability of 10 or more people out of 30000 going into a coma. Obviously, we don’t have a binomial table that will do this. Show that this is an appropriate place to use a Poisson distribution and use the Poisson tables to find the probability that On the basis of your result; do you think that something strange is going on? (The answer is shorter than the problem.)
g. (Extra credit) A telemarketer finds that the length of a call has an exponential distribution with a mean of minutes. Find the probability that the length of a call will be (i) No more than 3 minutes, (ii) Between 1 and 2 minutes, (iii) More than 5 minutes.
j. A coin is tossed 6 times. Define the following events.
(i) Find - Use a table in all three parts of this problem. (ii) Find . (iii) Find
1