Math Problem Statement

The histogram shows the distribution of hurricanes that have hit a country from 1851 through 2015, where 1 is the weakest level and 5 is the strongest level. (a) Find the mean, variance, and standard deviation of the probability distribution. (b) Interpret the results.

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: X={1,2,3,4,5}X = \{1, 2, 3, 4, 5\}
  • Corresponding probabilities: P(X)={0.411,0.257,0.212,0.109,0.011}P(X) = \{0.411, 0.257, 0.212, 0.109, 0.011\}

1. Mean (Already Given):

The mean (μ\mu) is already calculated and provided as 2.05.

2. Variance Formula:

The variance (σ2\sigma^2) is given by: σ2=P(X)(Xμ)2\sigma^2 = \sum P(X) \cdot (X - \mu)^2

Let's calculate it step by step:

  • For X=1X = 1: P(X=1)=0.411,(Xμ)2=(12.05)2=1.1025P(X=1) = 0.411, \quad (X - \mu)^2 = (1 - 2.05)^2 = 1.1025 P(X=1)(Xμ)2=0.4111.1025=0.4534275P(X=1) \cdot (X - \mu)^2 = 0.411 \cdot 1.1025 = 0.4534275

  • For X=2X = 2: P(X=2)=0.257,(Xμ)2=(22.05)2=0.0025P(X=2) = 0.257, \quad (X - \mu)^2 = (2 - 2.05)^2 = 0.0025 P(X=2)(Xμ)2=0.2570.0025=0.0006425P(X=2) \cdot (X - \mu)^2 = 0.257 \cdot 0.0025 = 0.0006425

  • For X=3X = 3: P(X=3)=0.212,(Xμ)2=(32.05)2=0.9025P(X=3) = 0.212, \quad (X - \mu)^2 = (3 - 2.05)^2 = 0.9025 P(X=3)(Xμ)2=0.2120.9025=0.19133P(X=3) \cdot (X - \mu)^2 = 0.212 \cdot 0.9025 = 0.19133

  • For X=4X = 4: P(X=4)=0.109,(Xμ)2=(42.05)2=3.8025P(X=4) = 0.109, \quad (X - \mu)^2 = (4 - 2.05)^2 = 3.8025 P(X=4)(Xμ)2=0.1093.8025=0.4148725P(X=4) \cdot (X - \mu)^2 = 0.109 \cdot 3.8025 = 0.4148725

  • For X=5X = 5: P(X=5)=0.011,(Xμ)2=(52.05)2=8.7025P(X=5) = 0.011, \quad (X - \mu)^2 = (5 - 2.05)^2 = 8.7025 P(X=5)(Xμ)2=0.0118.7025=0.0957275P(X=5) \cdot (X - \mu)^2 = 0.011 \cdot 8.7025 = 0.0957275

3. Variance Calculation:

σ2=0.4534275+0.0006425+0.19133+0.4148725+0.0957275=1.155999\sigma^2 = 0.4534275 + 0.0006425 + 0.19133 + 0.4148725 + 0.0957275 = 1.155999 So, the variance is approximately 1.16.

4. Standard Deviation:

The standard deviation (σ\sigma) is the square root of the variance: σ=1.1559991.075\sigma = \sqrt{1.155999} \approx 1.075 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:

  1. How would the variance and standard deviation change if the distribution had a higher probability in the category 4 range?
  2. What is the relationship between variance and standard deviation?
  3. Can the mean ever be outside the range of categories (1 to 5) in this context?
  4. How does the histogram's skew affect the mean and standard deviation?
  5. 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)