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Chapter 5: Problem 2

During a recent 64 -year period, New Mexico had a total of 153 tornadoes that measured 1 or greater on the Fujita scale. Let the random variable \(x\) represent the number of such tornadoes to hit New Mexico in one year, and assume that it has a Poisson distribution. What is the mean number of such New Mexico tornadoes in one year? What is the standard deviation? What is the variance?

Short Answer

Expert verified
The mean number of tornadoes per year is approximately 2.39. The standard deviation is approximately 1.55. The variance is 2.39.

Step by step solution

01

- Identify the data given

New Mexico experienced 153 tornadoes over a 64-year period. The random variable mentioned, x, represents the number of tornadoes in one year.
02

- Calculate the mean number of tornadoes per year

To find the mean, use the formula for the Poisson distribution's mean: \[ \text{Mean } (\text{渭}) = \frac{\text{Total number of events (k)}}{\text{Total period of time (t)}} \] Here, k = 153 and t = 64: \[ \text{渭} = \frac{153}{64} \text{ } \text{ or roughly } 2.39 \] So, the mean number of tornadoes per year is approximately 2.39.
03

- Understand variance and standard deviation for Poisson distribution

In a Poisson distribution, the mean (渭) is equal to the variance (蟽虏). Therefore, the variance for this distribution is also 2.39. For the standard deviation, it is the square root of the variance.
04

- Calculate the standard deviation

The standard deviation (蟽) is given by: \[ \text{蟽} = \text{鈭毾兟瞹 \] So, plugging in the variance: \[ \text{蟽} = \text{鈭�2.39} \text{ which is roughly } 1.55 \]

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Key Concepts

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

Mean Calculation
In statistics, the mean is a measure of the central tendency of a set of numbers. For a Poisson distribution, the mean \( \text{渭} \) tells us the average number of times the event occurs in a given time period. In this problem, New Mexico experienced 153 tornadoes over 64 years. To find the mean number of tornadoes per year, use the formula: \[ \text{Mean } (\text{渭}) = \frac{\text{Total number of events (k)}}{\text{Total period of time (t)}} \].
Here, \( k = 153 \) and \( t = 64 \). Calculating gives us: \[ \text{渭} = \frac{153}{64} \text{ which is approximately } 2.39 \].
So, the mean number of tornadoes per year in New Mexico is roughly 2.39.
Variance
Variance measures how much the values in a dataset differ from the mean. For a Poisson distribution, the variance \( 蟽^2 \) is equal to the mean. Thus, if the mean number of tornadoes per year is 2.39, the variance is also 2.39. This property helps in simplifying calculations. It emphasizes that the spread of tornado occurrences around the mean in New Mexico is characterized by the same value as the mean itself. Variance is crucial for understanding how much variability exists within the dataset.
Standard Deviation
Standard deviation is the square root of the variance and provides us with a measure of the dispersion of data points around the mean. It is particularly useful because it is in the same units as the data, making it easier to interpret. From our variance of 2.39, the standard deviation (蟽) is calculated by: \[ \text{蟽} = \text{鈭毾兟瞹 = \text{鈭�2.39} \]
This results in a standard deviation of approximately 1.55. Hence, the number of tornadoes per year deviates from the mean by about 1.55 tornadoes on average.
Tornado Statistics
Tornado statistics help meteorologists and planners understand and prepare for natural disasters. Based on historical data, one can derive important statistical measurements like the mean number of tornadoes, their variance, and standard deviation. These metrics play a critical role in risk assessment and resource allocation for emergency responses. For New Mexico, knowing that the mean number of tornadoes per year is 2.39 helps to shape preparedness strategies. Additionally, these statistics highlight the variability and likelihood of tornado occurrences over time.
Random Variable
A random variable represents a numerical outcome of a random phenomenon. In this exercise, the random variable \( x \) represents the number of tornadoes hitting New Mexico in one year. This variable follows a Poisson distribution, as the events (tornadoes) occur independently over a continuous interval (years). By modeling tornado occurrences with a Poisson distribution, we can effectively predict the probability of varying numbers of tornadoes in a given year. Understanding this concept is key for interpreting the statistical measures we've calculated.

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Most popular questions from this chapter

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