Ch. 4 - Discrete Probability Distributions

Larson - Elementary Statistics: Picturing the World 8th Edition

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Larson 8th Edition

Ch. 4 - Discrete Probability Distributions

Problem 4.T.2b

Chapter 4, Problem 4.T.2b

In Exercises 1–7, consider a grocery store that can process a total of four customers at its checkout counters each minute.
Minitab was used to generate 20 random numbers with a Poisson distribution for . Let the random number represent the number of arrivals at the checkout counter each minute for 20 minutes. 3 3 3 3 5 5 6 7 3 6 3 5 6 3 4 6 2 2 4 1During each of the first four minutes, only three customers arrived. These customers could all be processed, so there were no customers waiting after four minutes.
b. Create a table that shows the number of customers waiting at the end of 1 through 20 minutes.

Verified step by step guidance

Step 1: Understand the problem. The grocery store can process up to 4 customers per minute. If more than 4 customers arrive in a given minute, the excess customers will have to wait. The goal is to create a table showing the number of customers waiting at the end of each minute for 20 minutes.

Step 2: Define the variables. Let the number of arrivals each minute be represented by the given Poisson-distributed random numbers. Let the number of customers processed per minute be fixed at 4. The number of customers waiting at the end of each minute can be calculated as the cumulative excess arrivals minus the cumulative processing capacity.

Step 3: Calculate the excess arrivals for each minute. For each minute, subtract the processing capacity (4 customers) from the number of arrivals. If the result is positive, it represents the number of customers who cannot be processed and will wait. If the result is negative or zero, it means no customers are waiting.

Step 4: Compute the cumulative waiting customers. For each minute, add the excess arrivals from the current minute to the cumulative waiting customers from the previous minute. This will give the total number of customers waiting at the end of each minute.

Step 5: Create the table. Organize the results into a table with two columns: one for the minute (1 through 20) and one for the number of customers waiting at the end of that minute. Use the calculations from Steps 3 and 4 to populate the table.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Poisson Distribution

The Poisson distribution is a probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space, given that these events occur with a known constant mean rate and independently of the time since the last event. It is commonly used in scenarios where events happen randomly, such as customer arrivals at a store.

Queueing Theory

Queueing theory is the mathematical study of waiting lines or queues. It helps analyze the behavior of queues in various systems, including how many customers arrive, how long they wait, and how many can be served. In this context, it is essential for understanding customer flow and service efficiency at the grocery store's checkout counters.

Random Number Generation

Random number generation is the process of creating a sequence of numbers that lack any predictable pattern. In statistics, it is often used to simulate real-world processes, such as customer arrivals in this scenario. The generated random numbers can represent various outcomes, allowing for analysis and modeling of different situations.

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Related Practice

Textbook Question

The Centers for Disease Control and Prevention (CDC) is required by law to publish a report on assisted reproductive technology (ART). ART includes all fertility treatments in which both the egg and the sperm are used. These procedures generally involve removing eggs from a patient’s ovaries, combining them with sperm in the laboratory, and returning them to the patient’s body or giving them to another patient.

You are helping to prepare a CDC report on young ART patients and select at random 6 ART cycles of patients under 35 years of age for a special review. None of the cycles resulted in a live birth. Your manager feels it is impossible to select at random 10 ART cycles that do not result in a live birth. Use the pie chart at the right and your knowledge of statistics to determine whether your manager is correct.

a. How would you determine whether your manager is correct, that it is impossible to select at random six ART cycles that do not result in a live birth?

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Textbook Question

In Exercises 1–3, find the indicated probabilities using the geometric distribution, the Poisson distribution, or the binomial distribution. Then determine whether the events are unusual. If convenient, use a table or technology to find the probabilities.

One out of every 42 tax returns for incomes over \$1 million requires an audit. An auditor is examining tax returns for over \$1 million. Find the probability that (b) the first return requiring an audit is the first or second return the tax auditor examines,

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Textbook Question

In Exercises 1–7, consider a grocery store that can process a total of four customers at its checkout counters each minute.

The mean number of arrivals per minute is four. Find the probability that

b. more than four customers will arrive during the first minute.

Textbook Question

In Exercises 13–16, find the indicated binomial probabilities. If convenient, use technology or Table 2 in Appendix B.

Fifty-three percent of U.S. adults support attempting to land an astronaut on Mars. You randomly select eight U.S. adults. Find the probability that the number who support attempting to land an astronaut on Mars is (a) exactly three

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Textbook Question

In Exercises 1–7, consider a grocery store that can process a total of four customers at its checkout counters each minute.

The mean number of arrivals per minute is four. Find the probability that

c. more than four customers will arrive during each of the first four minutes.

Textbook Question

b. What probability distribution do you think best describes the situation? Do you think the distribution of the number of live births is discrete or continuous? Explain your reasoning.