91Ó°ÊÓ

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 20 per hour. The manager observes that 9 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 \(\mathrm{b}\), 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.

Step by step solution

01

Compute the Expected Number of Calls in 15 Minutes

First, the expected mean number of calls (\( \lambda \)) for a 15-minute duration is derived from the given hourly rates. This is done since the observation is for a 15-minute period. So, if the rate is 20 calls per hour, then \( \lambda = 20 * (15/60) = 5 \) calls for 15 minutes. If the rate is 30 calls per hour, then \( \lambda = 30 * (15/60) = 7.5 \) calls for 15 minutes.

02

Compute the Probability for 9 or More Calls

The formula to compute the Poisson probability for x occurrences is given by \( P(x; \lambda) = \frac{e^{-\lambda} * \lambda^x}{x!} \). However, we want to find the probability for 9 or more calls. That is 1 minus the sum of probabilities from 0 to 8 which can be calculated using Poisson Cumulative Distribution Function (CDF). Using the above formula, compute the probabilities for 20 and 30 calls per hour.

03

Compare the Probabilities

Comparison of the computed probabilities for 9 or more calls when the call rate is 20 or 30 calls per hour will aid in deciding which call rate is more likely, considering the observed data.

04

Provide Advice on Hiring

Based on the results obtained in Step 3, suggest whether the manager should consider hiring a second operator. This advice will be given considering the threshold set by the manager, which is more than 20 calls per hour.

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91Ó°ÊÓ!