Sure, let's break this down step by step:
Part (a): Finding the Probability
We need to find the probability that there will be exactly 5 hurricanes in a year, given that the average number of hurricanes per year is 6.1. This is a typical problem where we use the Poisson distribution. The Poisson probability formula is:
[tex]\[ P(X = k) = \frac{e^{-\lambda} \cdot \lambda^k}{k!} \][/tex]
Where:
- [tex]\( \lambda \)[/tex] (lambda) is the average rate (mean number of hurricanes per year = 6.1).
- [tex]\( k \)[/tex] is the actual number of hurricanes we want to find the probability for (k = 5).
- [tex]\( e \)[/tex] is Euler's number (approximately 2.71828).
Using this formula:
[tex]\[ P(X = 5) = \frac{e^{-6.1} \cdot 6.1^5}{5!} \][/tex]
After calculating the value, we find that the probability is approximately:
[tex]\[ P(X = 5) \approx 0.158 \][/tex]
So, the probability that there will be exactly 5 hurricanes in a year is 0.158 (rounded to three decimal places).
Part (b): Expected Number of Years
Next, we need to find the expected number of years with exactly 5 hurricanes over a 55-year period. This can be calculated by multiplying the probability found in part (a) by 55:
[tex]\[ \text{Expected Number of Years} = 55 \times 0.158 \][/tex]
Performing the multiplication:
[tex]\[ \text{Expected Number of Years} \approx 8.682 \][/tex]
So, over a 55-year period, we expect about 8.682 years to have exactly 5 hurricanes.
Part (c): Comparison with Observed Data
We now compare the expected number of years with the observed number of years. We have a recent period of 55 years in which 8 years had exactly 5 hurricanes.
- Expected number of years: 8.682
- Observed number of years: 8
To compare, we look at the difference between the observed and expected values:
[tex]\[ \text{Difference} = |8.682 - 8| \approx 0.682 \][/tex]
The difference between the expected number and the observed number is approximately 0.682, which is quite small. This suggests that the Poisson distribution works reasonably well for modeling the number of hurricanes in this scenario. The observed data closely matches the expected data derived from the Poisson distribution, indicating that this statistical model is appropriate for this context.
