SECTION 5.2 Exercises

For Exercises 5.40 and 5.41, see page 269; and for 5.42 and 5.43, see page 271.

Unless stated otherwise in the exercise, use software to find the exact Poisson.

Question 5.44

5.44 How many calls?

Calls to the customer service department of a cable TV provider are made randomly and independently at a rate of 11 per minute. The company has a staff of 20 customer service specialists who handle all the calls. Assume that none of the specialists are on a call at this moment and that a Poisson model is appropriate for the number of incoming calls per minute.

  1. What is the probability of the customer service department receiving more than 20 calls in the next minute?
  2. What is the probability of the customer service department receiving exactly 20 calls in the next minute?
  3. What is the probability of the customer service department receiving fewer than 11 calls in the next minute?

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Question 5.45

5.45 EPL goals.

Refer to Example 5.20 (pages 271–272) in which we found that the total number of goals scored in a game is well modeled by the Poisson distribution. Compute the following probabilities without the aid of software.

  1. What is the probability that a game will end in a 0–0 tie?
  2. What is the probability that three or more goals are scored in a game?

5.45

(a) 0.0628. (b) 0.5229.

Question 5.46

5.46 Email.

Suppose the average number of emails received by a particular employee at your company is five emails per hour. Suppose the count of emails received can be adequately modeled as a Poisson random variable. Compute the following probabilities without the aid of software.

  1. What is the probability of this employee receiving exactly five emails in any given hour?
  2. What is the probability of receiving less than five emails in any given hour?
  3. What is the probability of receiving at least one email in any given hour?
  4. What is the probability of receiving at least one email in any given 30-minute span?

Question 5.47

5.47 Traffic model.

The number of vehicles passing a particular mile marker during 15-minute units of time can be modeled as a Poisson random variable. Counting devices show that the average number of vehicles passing the mile marker every 15 minutes is 48.7.

  1. What is the probability of 50 or more vehicles passing the marker during a 15-minute time period?
  2. What is the standard deviation of the number of vehicles passing the marker in a 15-minute time period? A 30-minute time period?
  3. What is the probability of 100 or more vehicles passing the marker during a 30-minute time period?

5.47

(a) 0.4450 (0.4247 using Normal). (b) . . (c) 0.4095 (0.3974 using Normal).

Question 5.48

5.48 Flaws in carpets.

Flaws in carpet material follow the Poisson model with mean 0.8 flaw per square yard. Suppose an inspector examines a sample of carpeting measuring 1.25 yards by 1.5 yards.

  1. What is the distribution for the number of flaws in the sample carpeting?
  2. What is the probability that the total number of flaws the inspector finds is exactly five?
  3. What is the probability that the total number of flaws the inspector finds is two or less?

Question 5.49

5.49 Email, continued.

Refer to Exercise 5.46, where we learned that a particular employee at your company receives an average of five emails per hour.

  1. What is the distribution of the number of emails over the course of an eight-hour day?
  2. What is the probability of receiving 50 or more emails during an eight-hour day?

5.49

(a) Poisson with . (b) 0.0703 (0.0571 using Normal).

Question 5.50

5.50 Initial public offerings.

The number of companies making their initial public offering of stock (IPO) can be modeled by a Poisson distribution with a mean of 15 per month.

  1. What is the probability of three or fewer IPOs in a month?
  2. What is the probability of 10 or fewer in a two-month period?
  3. What is the probability of 200 or more IPOs in a year?
  4. Redo part (c) using the Normal approximation.

Question 5.51

5.51 How many zeroes expected?

Refer to Example 5.21 (page 272). We would find 1099 of the observed counts to have a value of 0. Based on the provided information in the example, how many more observed zeroes are there in the data set than what the best-fitting Poisson model would expect?

5.51

184.

Question 5.52

5.52 Website hits.

A “hit” for a website is a request for a file from the website’s server computer. Some popular websites have thousands of hits per minute. One popular website boasts an average of 6500 hits per minute between the hours of 9 A.M. and 6 P.M. Assume that the hits per hour are well modeled by the Poisson distribution. Some software packages will have trouble calculating Poisson probabilities with such a large value of .

  1. Use Excel’s Poisson function to calculate the probability of 6400 or more hits during the minute beginning at 10:05 A.M. What did you get?
  2. Find the probability of part (a) using the Normal approximation.
  3. Minitab users only: Try calculating the probability of part (a) using the Minitab’s Poisson option. Did you get an answer? If not, how did the software respond? What is the largest value of that Minitab can handle?

Question 5.53

5.53 Website hits, continued.

Refer to the previous exercise to determine the number of website hits in one hour. Use the Normal distribution to find the range in which we would expect 99.7% of the hits to fall.

5.53

Between 388126.5 and 391873.5.

Question 5.54

5.54 Mishandled baggage.

In the airline industry, the term “mishandled baggage” refers to baggage that was lost, delayed, damaged, or stolen. In 2013, American Airlines had an average of 3.02 mishandled baggage per 1000 passengers.14 Consider an incoming American Airlines flight carrying 400 passengers. Let be the number of mishandled baggage.

  1. Use the binomial distribution to find the probability that there will be at least one mishandled piece of baggage.
  2. Use the Normal approximation with continuity correction to find the probability of part (a).
  3. Use the Poisson approximation to find the probability of part (a).
  4. Which approximation was closer to the exact value? Explain why this is the case.

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Question 5.55

5.55 Calculator convenience.

Suppose that follows a Poisson distribution with mean .

  1. Show that .
  2. Show that for any whole number .
  3. Suppose . Use part (a) to compute .
  4. Part (b) gives us a nice calculator convenience that allows us to multiply a given Poisson probability by a factor to get the next Poisson probability. What would you multiply the probability from part (c) by to get ? What would you then multiply by to get ?

5.55

(a) .
(b)
(c) .
(d) .

Question 5.56

5.56 Baseball runs scored.

We found in Example 5.20 (pages 271–272) that, in soccer, goal scoring is well described by the Poisson model. It will be interesting to investigate if that phenomenon carries over to other sports. Consider data on the number of runs scored per game by the Washington Nationals for the 2013 season. Parts (a) through (e) can be done with any software.

washnat

  1. Produce a histogram of the runs. Describe the distribution.
  2. What is the mean number of runs scored by the Nationals? What is the sample variance of the runs scored?
  3. What do your answers from part (b) tell you about the applicability of the Poisson model for these data?
  4. If you were to use the Poisson model, how many games in a 162-game season would you expect the Nationals not to score in?
  5. Sort the runs scored column and count the actual number of games that the Nationals did not score in. Compare this count with part (d) and respond.
  6. JMP users only: Provide output of Poisson fit superimposed on the histogram of runs. To do this, first create a histogram using the Distribution platform and then pick the Poisson option found in the Discrete Fit option. Discuss what you see.