Question

In: Statistics and Probability

The number of calls that come into a small mail-order company follows a Poisson distribution. Currently,...

The number of calls that come into a small mail-order company follows a Poisson distribution. Currently, these calls are serviced by a single operator. The manager knows from past experience that an additional operator will be needed if the rate of calls exceeds 15 per hour. The manager observes that 6 calls came into the mail-order company during a randomly selected 15-minute period.

a. If the rate of calls is actually 20 per hour, what is the probability that 9 or more calls will come in during a given 15-minute period?

b. If the rate of calls is really 30 per hour, what is the probability that 9 or more calls will come in during a given 15-minute period?

c. Based on the calculations in parts a and b, do you think that the rate of incoming calls is more likely to be 20 or 30 per hour?

d. Would you advise the manager to hire a second operator ? Explain.

Solutions

Expert Solution

POISSON DISTRIBUTION (linear adjustment)

a) You have a random variable X(the number of calls which come in service mail order ), which has Poisson distribution, with parameter λ=20. The probability that there will be k "events" in time t, where tis measured in hours, is

e-20t (20t)k/k!  

Let Y be a new random variable, the number of events in time s, where s is time measured in units of 15/60 = 1/4hour(for 15minutes). I changed the letter used for time, so as to avoid confusion with time as measured in hours. And I am using a new letter for the random variable, to prevent confusion with X

The random variable Y has Poisson distribution, with parameter θ=20/4=5, In other words, the parameter scales as one would expect. After all, for the Poisson, the parameter is the mean.

Then for example the probability that there will be k events in time interval of length s=1(quarter-hours) is given by

e-5(5 )k/ k!

(We did not need to invent the new Y to find this probability. In the original X, we could simply let t=1/4t=1/4.)

The probability of k events in time interval s=4s=4 (aka 1 hour) is also easily computed. Here θs=4θs=20, and you get probability

e-20t (20t)k/k!  

now let's compute the probability

P(X>=9 ) = = 1-(P(X=8)+P(X=7)+....P(X=0) =  0.06810 ~ 6.8%

B) AVG RATE OF INCOMING CALLS = 30 PER HOUR

30 PER HOUR = 30/60 PER MINUTE = 1/2 PER MINUTE * 15 = 15/2 PER 15 MINUTES

P(X>=9) =. =   1-(P(X=8)+P(X=7)+....P(X=0) = 0.33803~33.803%

c) avg rate of incoming calls can be 20 per hours or 30 per hour as the observed rate of incoming calls in random 15minutes by manager was 6 per 15min or 24 per hour which lies in this interval , so yes incoming call rate can be in between but we cannot conclude it on the basis of one random 15 min observation , we need more observations to explore it further

d) i would suggest manager to hire a second operator as the rate of incoming can be more than 15calls per hour but we cannot conclude it because of only 1 random 15 minute observation , we need more observations like this to explore it further


Related Solutions

The number of calls that come into a small mail-order company follows a Poisson distribution. Currently,...
The number of calls that come into a small mail-order company follows a Poisson distribution. Currently, these calls are serviced by a single operator. The manager knows from past experience that an additional operator will be needed if the rate of calls exceeds 15 per hour. The manager observes that 6 calls came into the mail-order company during a randomly selected 15-minute period. a. If the rate of calls is actually 20 per hour, what is the probability that 9...
The number of airplane landings at a small airport follows a Poisson distribution with a mean...
The number of airplane landings at a small airport follows a Poisson distribution with a mean rate of 3 landings every hour. 4.1 (6%) Compute the probability that 3 airplanes arrive in a given hour? 4.2 (7%) What is the probability that 8 airplanes arrive in three hours? 4.3 (7%) What is the probability that more than 3 airplanes arrive in a period of two hours?
The number of complaints received by a company each month follows a Poisson distribution with mean...
The number of complaints received by a company each month follows a Poisson distribution with mean 6. (a) Calculate the probability the company receives no complaint in a certain week. (b) Calculate the probability the company receives more than 4 complaints in a 2-week period. (c) Over a certain month, calculate the probability the company receives fewer complaints than it usually does with respect to its monthly average.
The number of views of a page on a Web site follows a Poisson distribution with...
The number of views of a page on a Web site follows a Poisson distribution with a mean of 1.5 per minute. (i) What is the probability of no views in a minute? (ii) What is the probability of two or fewer views in 10 minutes? (iii) What is the probability of two or fewer views in 2 hours?
Suppose the number of earthquakes occurring in an area approximately follows a Poisson distribution with an...
Suppose the number of earthquakes occurring in an area approximately follows a Poisson distribution with an average rate of 2 earthquakes every year. a.) Find the probability that there will be 1 to 3 (inclusive) earthquakes during the next year in this area. b.) Find the probability that there will be exactly 5 earthquakes during the next 3 year period. c.) Consider 10 randomly selected years during last century. What is the probability that there will be at least 3...
The number of people arriving at an emergency room follows a Poisson distribution with a rate...
The number of people arriving at an emergency room follows a Poisson distribution with a rate of 9 people per hour. a) What is the probability that exactly 7 patients will arrive during the next hour? b. What is the probability that at least 7 patients will arrive during the next hour? c. How many people do you expect to arrive in the next two hours? d. One in four patients who come to the emergency room in hospital. Calculate...
The number of people crossing at a certain traffic light follows a Poisson distribution with a...
The number of people crossing at a certain traffic light follows a Poisson distribution with a mean of 6 people per hour. An investigator is planning to count the number of people crossing the light between 8pm and 10pm on a randomly selected 192 days. Let Y be the average of the numbers of people to be recorded by the investigator. What is the value of E[Y ]? A. 6 B. 12 C. 1152 D. 0.03125. What is the value...
The number of people arriving at an emergency room follows a Poisson distribution with a rate...
The number of people arriving at an emergency room follows a Poisson distribution with a rate of 7 people per hour. a. What is the probability that exactly 5 patients will arrive during the next hour? (3pts) b. What is the probability that at least 5 patients will arrive during the next hour? (5pts) c. How many people do you expect to arrive in the next two hours? (4pts) d. One in four patients who come to the emergency room...
The number of earthquakes that occur per week in California follows a Poisson distribution with a...
The number of earthquakes that occur per week in California follows a Poisson distribution with a mean of 1.5. (a) What is the probability that an earthquake occurs within the first week? Show by hand and provide the appropriate R code. (b) What is the expected amount of time until an earthquake occurs? (c) What is the standard deviation of the amount of time until two earthquakes occur? (d) What is the probability that it takes more than a month...
The number of orders that come into a mail-order sales office each month is normally distributed...
The number of orders that come into a mail-order sales office each month is normally distributed with a population mean of 298 and a population standard deviation of 15.4. For a particular sample size, the probability is 0.2 that the sample mean exceeds 300. How big must the sample be?
ADVERTISEMENT
ADVERTISEMENT
ADVERTISEMENT