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.5

Chapter 4, Problem 4.T.5

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 customers who arrive at the checkout counters each minute is 4. Create a Poisson distribution with mu = 4 for x = 0 to 20. Compare your results with the histogram shown at the upper right.

Verified step by step guidance

Understand the problem: The Poisson distribution is used to model the number of events (in this case, customer arrivals) occurring in a fixed interval of time. The mean (μ) is given as 4, and we are tasked with creating a Poisson distribution for x = 0 to 20.

Recall the formula for the Poisson probability mass function (PMF): P(X = x) = (e^(-μ) * μ^x) / x!, where μ is the mean, x is the number of events, and e is the base of the natural logarithm (approximately 2.718).

For each value of x from 0 to 20, calculate the probability using the Poisson PMF formula. For example, for x = 0, P(X = 0) = (e^(-4) * 4^0) / 0!. Repeat this calculation for x = 1, x = 2, ..., up to x = 20.

Once the probabilities are calculated, create a table or list of the values of x (from 0 to 20) and their corresponding probabilities. This will represent the Poisson distribution for the given mean μ = 4.

To compare your results with the histogram, plot the calculated probabilities as a bar graph (histogram) with x on the horizontal axis and P(X = x) on the vertical axis. Visually compare the shape and distribution of your graph with the provided histogram.

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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 a known average rate of occurrence. It is characterized by the parameter 'mu' (λ), which represents the average number of events in the interval. In this case, with mu = 4, it models the number of customers arriving at the checkout counters per minute.

Mean (Expected Value)

The mean, or expected value, of a probability distribution is the long-term average value of repetitions of the experiment it represents. For the Poisson distribution, the mean is equal to the parameter mu (λ). In this scenario, a mean of 4 indicates that, on average, 4 customers arrive at the checkout counters each minute.

Histogram

A histogram is a graphical representation of the distribution of numerical data, where the data is divided into bins or intervals. Each bin's height represents the frequency of data points within that interval. In this context, comparing the histogram of the Poisson distribution with the calculated probabilities helps visualize how the number of customers arriving aligns with the expected distribution.

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Textbook Question

Determine whether the distribution is a probability distribution. If it is not a probability distribution, explain why.

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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 increases to five arrivals per minute, but the store can still process only four per minute. Generate a list of 20 random numbers with a Poisson distribution for mu = 5 . Then create a table that shows the number of customers waiting at the end of 20 minutes.

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Textbook Question

Using a Distribution to Find Probabilities In Exercises 11–26, 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.

Oil Tankers In the month of June 2021, 240 oil tankers stop at a port city. No oil tanker visits more than once. Find the probability that the number of oil tankers that stop on any given day in June is (a) exactly eight

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Textbook Question

Hypergeometric Distribution Binomial experiments require that any sampling be done with replacement because each trial must be independent of the others. The hypergeometric distribution also has two outcomes: success and failure. The sampling, however, is done without replacement. For a population of N items having k successes and failures, the probability of selecting a sample of size that has successes and failures is given by

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In a shipment of 15 microchips, 2 are defective and 13 are not defective. A sample of three microchips is chosen at random. Use the above formula to find the probability that (a) all three microchips are not defective

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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. one customer is waiting in line after one minute and no customers are waiting in line after the second minute..

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Textbook Question

Finding Probabilities Use the probability distribution you made in Exercise 19 to find the probability of randomly selecting a household that has (a) one or two HD televisions

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