Textbook Question
In Exercises 9–16, use the Poisson distribution to find the indicated probabilities.
Births In a recent year (365 days), NYU-Langone Medical Center had 5942 births.
a. Find the mean number of births per day.
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Ch. 5 - Discrete Probability Distributions
Triola14th EditionElementary StatisticsISBN: 9780137366446
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Table of contents
- Ch. 1 - Introduction to Statistics88
- Ch. 2 - Exploring Data with Tables and Graphs70
- Ch. 3 - Describing, Exploring, and Comparing Data66
- Ch. 4 - Probability112
- Ch. 5 - Discrete Probability Distributions109
- Ch. 6 - Normal Probability Distributions122
- Ch. 7 - Estimating Parameters and Determining Sample Sizes107
- Ch. 8 - Hypothesis Testing75
- Ch. 9 - Inferences from Two Samples94
- Ch. 10 - Correlation and Regression80
- Ch. 11 - Goodness-of-Fit and Contingency Tables29
- Ch. 12 - Analysis of Variance42
Chapter 5, Problem 5.3.8a
In Exercises 5–8, assume that the Poisson distribution applies; assume that the mean number of Atlantic hurricanes in the United States is 5.5 per year, as in Example 1; and proceed to find the indicated probability.
a. Find the probability that in a year, there will be 10 hurricanes.
Verified step by step guidance1
Understand the problem: The Poisson distribution is used to model the number of events (hurricanes) occurring in a fixed interval of time (a year) when the events occur independently and at a constant average rate. Here, the mean number of hurricanes per year (λ) is 5.5, and we are tasked with finding the probability of observing exactly 10 hurricanes in a year.
Recall the formula for the Poisson probability mass function (PMF): P(X = k) = (λ^k * e^(-λ)) / k!, where λ is the mean number of events, k is the number of events we are interested in, and e is the base of the natural logarithm (approximately 2.718).
Substitute the given values into the formula: Set λ = 5.5 and k = 10. The formula becomes P(X = 10) = ((5.5)^10 * e^(-5.5)) / 10!.
Break down the calculation into manageable parts: (1) Compute (5.5)^10, which is the mean raised to the power of 10. (2) Compute e^(-5.5), which is the exponential function with a negative mean. (3) Compute 10!, which is the factorial of 10 (i.e., 10 × 9 × 8 × ... × 1).
Combine the results: Multiply (5.5)^10 by e^(-5.5), then divide the result by 10!. This will give you the probability of observing exactly 10 hurricanes in a year.
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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 particularly useful for modeling rare events, such as the number of hurricanes in a year, where the events are independent of each other.
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Intro to Frequency Distributions
Mean (λ) in Poisson Distribution
In the context of the Poisson distribution, the mean (denoted as λ, lambda) represents the average number of occurrences of the event in the specified interval. For this question, λ is given as 5.5, indicating that, on average, there are 5.5 hurricanes per year in the United States, which serves as a key parameter for calculating probabilities.
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Calculating Probability with Poisson Formula
To find the probability of observing a specific number of events (k) in a Poisson distribution, the formula P(X = k) = (e^(-λ) * λ^k) / k! is used, where e is Euler's number (approximately 2.71828), λ is the mean, and k! is the factorial of k. This formula allows us to compute the likelihood of experiencing exactly k events, such as 10 hurricanes in a year.
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Related Practice
Textbook Question
Lottery. In Exercises 15–20, refer to the accompanying table, which describes probabilities for the California Daily 4 lottery. The player selects four digits with repetition allowed, and the random variable x is the number of digits that match those in the same order that they are drawn (for a “straight” bet).
Using Probabilities for Significant Events
a. Find the probability of getting exactly 3 matches.
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Textbook Question
In Exercises 1–5, assume that 4.2% of workers test positive when tested for illegal drugs (based on data from Quest Diagnostics). Assume that a group of ten workers is randomly selected.
Workplace Drug Testing If four of the ten workers test positive for illegal drugs, is that a significantly high result?
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Textbook Question
Family/Partner Groups of people aged 15–65 are randomly selected and arranged in groups of six. The random variable x is the number in the group who say that their family and/or partner contribute most to their happiness (based on a Coca-Cola survey). The accompanying table lists the values of x along with their corresponding probabilities. Does the table describe a probability distribution? If so, find the mean and standard deviation.
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Textbook Question
Acrophobia USA Today reported results from a survey in which subjects were asked if they are afraid of heights in tall buildings. The results are summarized in the accompanying table. Does this table describe a probability distribution? Why or why not?
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Textbook Question
In Exercises 5–8, assume that the Poisson distribution applies; assume that the mean number of Atlantic hurricanes in the United States is 5.5 per year, as in Example 1; and proceed to find the indicated probability.
a. Find the probability that in a year, there will be no hurricanes.
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