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.
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 5.3.7a
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.
Verified step by step guidance1
Step 1: Recall the formula for the Poisson probability distribution: P(X = k) = (λ^k * e^(-λ)) / k!, where λ is the mean number of occurrences, k is the number of occurrences we are interested in, and e is the base of the natural logarithm (approximately 2.718).
Step 2: Identify the given values from the problem. Here, λ = 5.5 (the mean number of hurricanes per year) and k = 0 (since we are finding the probability of no hurricanes in a year).
Step 3: Substitute the values into the Poisson formula. This gives P(X = 0) = (5.5^0 * e^(-5.5)) / 0!.
Step 4: Simplify the terms in the formula. Note that any number raised to the power of 0 is 1, so 5.5^0 = 1. Also, 0! (zero factorial) is equal to 1. Therefore, the formula simplifies to P(X = 0) = (1 * e^(-5.5)) / 1.
Step 5: The final step is to compute e^(-5.5) and divide by 1 to find the probability. This step involves using a calculator or software to evaluate the exponential term.
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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.
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Calculating Probability with Poisson
To find the probability of observing a specific number of events in a Poisson distribution, the formula P(X=k) = (e^(-λ) * λ^k) / k! is used, where P(X=k) is the probability of k events occurring, e is Euler's number, and k! is the factorial of k. For this question, to find the probability of no hurricanes (k=0), you would substitute λ=5.5 and k=0 into the formula.
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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.
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 10 hurricanes.
Textbook Question
Binomial Probability Formula. In Exercises 13 and 14, answer the questions designed to help understand the rationale for the binomial probability formula.
Guessing Answers Standard tests, such as the SAT, ACT, or Medical College Admission Test (MCAT), typically use multiple choice questions, each with five possible answers (a, b, c, d, e), one of which is correct. Assume that you guess the answers to the first three questions.
a. Use the multiplication rule to find the probability that the first two guesses are wrong and the third is correct. That is, find P(WWC), where W denotes a wrong answer and C denotes a correct answer.
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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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