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Chapter 5: Problem 5

Assume that the Poisson distribution applies; assume that the mean number of Atlantic hurricanes in the United States is 6.1 per year, as in Example \(I\); and proceed to find the indicated probability. Hurricanes a. Find the probability that in a year, there will be 5 hurricanes. b. In a 55 -year period, how many years are expected to have 5 hurricanes? c. How does the result from part (b) compare to the recent period of 55 years in which 8 years had 5 hurricanes? Does the Poisson distribution work well here?

Short Answer

Expert verified
Find \( P(X = 5) = 0.1606 \. Expected years in 55 is 8.833. Comparing with 8 shows a fairly good fit.

Step by step solution

01

Understand the Poisson Distribution

The Poisson distribution is used to model the number of events (hurricanes, in this case) that occur in a fixed interval of time or space. It is defined by the mean number of events (denoted by \(\lambda\)) which is given as 6.1 hurricanes per year.
02

Define the Poisson Probability Formula

The probability of observing exactly \(k\) events (hurricanes) in a Poisson distribution is given by: \[ P(X = k) = \frac{\lambda^k \cdot e^{-\lambda}}{k!} \] Where \( \lambda = 6.1 \) and \( k = 5 \).
03

Calculate the Probability for part (a)

Using the Poisson probability formula, substitute \(\lambda = 6.1\) and \(k = 5\): \[ P(X = 5) = \frac{6.1^5 \cdot e^{-6.1}}{5!} \] Compute this value using a calculator.
04

Calculate part (b)

To find the expected number of years in a 55-year period with 5 hurricanes, multiply the probability from part (a) by 55: \[ E = 55 \cdot P(X=5) \].
05

Compare with Recent Data for part (c)

Compare the expected number from part (b) to the observed 8 years in the last 55 years. Determine if 8 is close to the expected value to assess if the Poisson distribution fits the observed data.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Probability Theory
Probability theory helps us understand and predict the likelihood of different events.
It uses mathematics to model situations where outcomes are uncertain.
In the context of hurricanes, probability theory can help us understand how likely it is to have a certain number of hurricanes in a year.
This is useful for planning and preparation.
One important concept in probability theory is that of a distribution, which describes how probabilities are spread out over different possible outcomes.
Mean Number of Events
The mean number of events, often denoted by the Greek letter \( \lambda \), is a measure of the average rate at which events happen.
For the Poisson distribution, \( \lambda \) is a key parameter.
In our hurricanes example, \( \lambda \) is 6.1, meaning on average, there are 6.1 hurricanes per year.
Knowing the mean number of events is crucial for calculating probabilities.
This is because the mean helps to determine the probability of having a certain specific number of events, such as our 5 hurricanes in a year.
Statistical Modeling
Statistical modeling involves using data to create a model that describes a particular phenomenon.
The Poisson distribution is one example of a statistical model.
We use it to model the number of times an event happens in a fixed interval, like the number of hurricanes in a year.
By fitting our data to a Poisson model, we can make predictions and decisions based on the likelihood of various outcomes.
Statistical models can be very powerful tools in many fields, from meteorology to economics.
Poisson Probability Formula
The Poisson probability formula helps us calculate the probability of observing a specific number of events in a fixed interval.
It is given by: \[ P(X = k) = \frac{ \lambda^k \cdot e^{-\lambda}}{k!} \]
In this formula, \( \lambda \) is the mean number of events, \( k \) is the number of events we are interested in, and \( e \) is a constant approximately equal to 2.71828.
This formula is used to calculate the probability of having exactly \( k \) hurricanes in a year, given a mean of \( \lambda = 6.1 \).
For example, to find the probability of having 5 hurricanes, we substitute \( \lambda = 6.1 \) and \( k = 5 \) into the formula and solve.

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Most popular questions from this chapter

Based on a Pitney Bowes survey, assume that \(42 \%\) of consumers are comfortable having drones deliver their purchases. Suppose we want to find the probability that when five consumers are randomly selected, exactly two of them are comfortable with the drones. What is wrong with using the multiplication rule to find the probability of getting two consumers comfortable with drones followed by three consumers not comfortable, as in this calculation: \((0.42)(0.42)(0.58)(0.58)(0.58)=0.0344 ?\)

Determine whether a probability distribution is given. If a probability distribution is given, find its mean and standard deviation. If a probability distribution is not given, identify the requirements that are not satisfied. Groups of adults are randomly selected and arranged in groups of three. The random variable \(x\) is the number in the group who say that they would feel comfortable in a selfdriving vehicle (based on a TE Connectivity survey). $$\begin{array}{|c|c|} \hline x & P(x) \\ \hline 0 & 0.358 \\ \hline 1 & 0.439 \\ \hline 2 & 0.179 \\ \hline 3 & 0.024 \\ \hline \end{array}$$

Assume that the Poisson distribution applies; assume that the mean number of Atlantic hurricanes in the United States is 6.1 per year, as in Example \(I\); and proceed to find the indicated probability. Hurricanes a. Find the probability that in a year, there will be 7 hurricanes. b. In a 55 -year period, how many years are expected to have 7 hurricanes? c. How does the result from part (b) compare to the recent period of 55 years in which 7 years had 7 hurricanes? Does the Poisson distribution work well here?

Refer to the accompanying table, which describes the numbers of adults in groups of five who reported sleepwalking (based on data from "Prevalence and Comorbidity of Nocturnal Wandering In the U.S. Adult General Population," by Ohayon et al., Neurology, Vol. 78, No. 20). $$\begin{array}{|c|c|} \hline x & P(x) \\ \hline 0 & 0.172 \\ \hline 1 & 0.363 \\ \hline 2 & 0.306 \\ \hline 3 & 0.129 \\ \hline 4 & 0.027 \\ \hline 5 & 0.002 \\ \hline \end{array}$$ a. Find the probability of getting exactly 1 sleepwalker among 5 adults. b. Find the probability of getting 1 or fewer sleepwalkers among 5 adults. c. Which probability is relevant for determining whether 1 is a significantly low number of sleepwalkers among 5 adults: the result from part (a) or part (b)? d. Is 1 a significantly low number of sleepwalkers among 5 adults? Why or why not?

Involve finding binomial probabilities, finding parameters, and determining whether values are significantly high or low by using the range rule of thumb and probabilities. In a presidential election, 611 randomly selected voters were surveyed, and 308 of them said that they voted for the winning candidate (based on data from ICR Survey Research Group). The actual percentage of votes for the winning candidate was \(43 \%\). Assume that \(43 \%\) of voters actually did vote for the winning candidate, and assume that 611 voters are randomly selected. a. Use the range rule of thumb to identify the limits separating values that are significantly low and those that are significantly high. Based on the results, is the 308 voters who said that they voted for the winner significantly high? b. Find the probability of exactly 308 voters who actually voted for the winner. c. Find the probability of 308 or more voters who actually voted for the winner. d. Which probability is relevant for determining whether the value of 308 voters is significantly high: the probability from part (b) or part (c)? Based on the relevant probability, is the result of 308 voters who said that they voted for the winner significantly high? e. What is an important observation about the survey results?
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