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3.136 3.137 nomoM e.c Increased research and discussion have focused on the number of illnesses involving the organ- ism Escherichia coli (10257:H7), which causes a breakdown of red blood cells and intestinal hemorrhages in its victims (http://www.hsus.org/ace/11831, March 24, 2004). Sporadic out- breaks of E.coli have occurred in Colorado at a rate of approximately 2.4 per 100,000 for a period of two years. of making at a If this rate has not changed and if 100,000 cases from Colorado are reviewed for this year, what is the probability that at least 5 cases of E.coli will be observed? b If 100,000 cases from Colorado are reviewed for this year and the number of E.coli cases exceeded 5, would you suspect that the state's mean E.coli rate has changed? Explain. The probability that Al

3.136 3.137 nomoM e.c Increased research and discussion have focused on the number of illnesses involving the organ- ism Escherichia coli (10257:H7), which causes a breakdown of red blood cells and intestinal hemorrhages in its victims (http://www.hsus.org/ace/11831, March 24, 2004). Sporadic out- breaks of E.coli have occurred in Colorado at a rate of approximately 2.4 per 100,000 for a period of two years. of making at a If this rate has not changed and if 100,000 cases from Colorado are reviewed for this year, what is the probability that at least 5 cases of E.coli will be observed? b If 100,000 cases from Colorado are reviewed for this year and the number of E.coli cases exceeded 5, would you suspect that the state's mean E.coli rate has changed? Explain. The probability that Al

A First Course in Probability (10th Edition)
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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Please how do I solve Question 3.136
3.133 3.134 3.135 3.136 3.137 3.138 3.139 *3.140 3.141 Exercises 137 situation. Find the probability that the number of autos entering the tunnel during a two-minute period exceeds three. Does the Poisson model seem reasonable for this problem? Assume that the tunnel in Exercise 3.132 is observed during ten two-minute intervals, thus giving ten independent observations Y₁, Y2,..., Y10, on the Poisson random variable. Find the probability that Y> 3 during at least one of the ten two-minute intervals. Consider a binomial experiment for n = 20, p= .05. Use Table 1, Appendix 3, to calculate the binomial probabilities for Y = 0, 1, 2, 3, and 4. Calculate the same probabilities by using the Poisson approximation with λ = np. Compare. A salesperson has found that the probability of a sale on a single contact is approximately .03. If the salesperson contacts 100 prospects, what is the approximate probability of making at least one sale? Increased research and discussion have focused on the number of illnesses involving the organ- ism Escherichia coli (10257:H7), which causes a breakdown of red blood cells and intestinal hemorrhages in its victims (http://www.hsus.org/ace/11831, March 24, 2004). Sporadic out- breaks of E.coli have occurred in Colorado at a rate of approximately 2.4 per 100,000 for a period of two years. a If this rate has not changed and if 100,000 cases from Colorado are reviewed for this year, what is the probability that at least 5 cases of E.coli will be observed? b If 100,000 cases from Colorado are reviewed for this year and the number of E.coli cases exceeded 5, would you suspect that the state's mean E.coli rate has changed? Explain. The probability that a mouse inoculated with a serum will contract a certain disease is .2. Using the Poisson approximation, find the probability that at most 3 of 30 inoculated mice will contract the disease. Let Y have a Poisson distribution with mean λ. Find E[Y(Y-1)] and then use this to show that V (Y) = λ. In the daily production of a certain kind of rope, the number of defects per foot Y is assumed to have a Poisson distribution with mean λ 2. The profit per foot when the rope is sold is given by X, where X = 50-2Y - Y². Find the expected profit per foot. A store owner has overstocked a certain item and decides to use the following promotion to decrease the supply. The item has a marked price of $100. For each customer purchasing the item during a particular day, the owner will reduce the price by a factor of one-half. Thus, the first customer will pay $50 for the item, the second will pay $25, and so on. Suppose that the number of customers who purchase the item during the day has a Poisson distribution with mean 2. Find the expected cost of the item at the end of the day. [Hint: The cost at the end of the day is 100(1/2), where Y is the number of customers who have purchased the item.] A food manufacturer uses an extruder (a machine that produces bite-size cookies and snack food) that yields revenue for the firm at a rate of $200 per hour when in operation. However, the extruder breaks down an average of two times every day it operates. If Y denotes the number of breakdowns per day, the daily revenue generated by the machine is R = 1600-50Y². Find | 11 for the extruder
Transcribed Image Text:3.133 3.134 3.135 3.136 3.137 3.138 3.139 *3.140 3.141 Exercises 137 situation. Find the probability that the number of autos entering the tunnel during a two-minute period exceeds three. Does the Poisson model seem reasonable for this problem? Assume that the tunnel in Exercise 3.132 is observed during ten two-minute intervals, thus giving ten independent observations Y₁, Y2,..., Y10, on the Poisson random variable. Find the probability that Y> 3 during at least one of the ten two-minute intervals. Consider a binomial experiment for n = 20, p= .05. Use Table 1, Appendix 3, to calculate the binomial probabilities for Y = 0, 1, 2, 3, and 4. Calculate the same probabilities by using the Poisson approximation with λ = np. Compare. A salesperson has found that the probability of a sale on a single contact is approximately .03. If the salesperson contacts 100 prospects, what is the approximate probability of making at least one sale? Increased research and discussion have focused on the number of illnesses involving the organ- ism Escherichia coli (10257:H7), which causes a breakdown of red blood cells and intestinal hemorrhages in its victims (http://www.hsus.org/ace/11831, March 24, 2004). Sporadic out- breaks of E.coli have occurred in Colorado at a rate of approximately 2.4 per 100,000 for a period of two years. a If this rate has not changed and if 100,000 cases from Colorado are reviewed for this year, what is the probability that at least 5 cases of E.coli will be observed? b If 100,000 cases from Colorado are reviewed for this year and the number of E.coli cases exceeded 5, would you suspect that the state's mean E.coli rate has changed? Explain. The probability that a mouse inoculated with a serum will contract a certain disease is .2. Using the Poisson approximation, find the probability that at most 3 of 30 inoculated mice will contract the disease. Let Y have a Poisson distribution with mean λ. Find E[Y(Y-1)] and then use this to show that V (Y) = λ. In the daily production of a certain kind of rope, the number of defects per foot Y is assumed to have a Poisson distribution with mean λ 2. The profit per foot when the rope is sold is given by X, where X = 50-2Y - Y². Find the expected profit per foot. A store owner has overstocked a certain item and decides to use the following promotion to decrease the supply. The item has a marked price of $100. For each customer purchasing the item during a particular day, the owner will reduce the price by a factor of one-half. Thus, the first customer will pay $50 for the item, the second will pay $25, and so on. Suppose that the number of customers who purchase the item during the day has a Poisson distribution with mean 2. Find the expected cost of the item at the end of the day. [Hint: The cost at the end of the day is 100(1/2), where Y is the number of customers who have purchased the item.] A food manufacturer uses an extruder (a machine that produces bite-size cookies and snack food) that yields revenue for the firm at a rate of $200 per hour when in operation. However, the extruder breaks down an average of two times every day it operates. If Y denotes the number of breakdowns per day, the daily revenue generated by the machine is R = 1600-50Y². Find | 11 for the extruder
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