Math Problem Statement
Solution
Let's work through the problem by first calculating the variance and standard deviation based on the given probability distribution.
The histogram provides the following information:
- Categories:
- Corresponding probabilities:
1. Mean (Already Given):
The mean () is already calculated and provided as 2.05.
2. Variance Formula:
The variance () is given by:
Let's calculate it step by step:
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For :
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For :
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For :
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For :
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For :
3. Variance Calculation:
So, the variance is approximately 1.16.
4. Standard Deviation:
The standard deviation () is the square root of the variance: So, the standard deviation is approximately 1.08.
5. Interpret the Results:
- Based on the values, the average hurricane is approximately category 2.
- The typical hurricane deviates from the mean by about 1 category level, as indicated by the standard deviation.
Do you need more details or clarification?
Here are 5 related questions for further practice:
- How would the variance and standard deviation change if the distribution had a higher probability in the category 4 range?
- What is the relationship between variance and standard deviation?
- Can the mean ever be outside the range of categories (1 to 5) in this context?
- How does the histogram's skew affect the mean and standard deviation?
- What is the significance of a lower or higher standard deviation in interpreting hurricane strength?
Tip: The standard deviation provides insight into how spread out the data is around the mean. The larger the standard deviation, the more variability there is.
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Math Problem Analysis
Mathematical Concepts
Probability Distribution
Mean
Variance
Standard Deviation
Formulas
Mean formula: μ = Σ [X * P(X)]
Variance formula: σ² = Σ [P(X) * (X - μ)²]
Standard Deviation: σ = √σ²
Theorems
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Suitable Grade Level
College Level (Statistics or Honors Probability and Statistics)
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