A team of ecologists laid 100 plots of 50 m \(\times\) 50 m in a forest and counted the number of individuals of a tree species in each plot. They then calculated the mean and variance of the number of individuals per plot. If trees are randomly distributed, then which one of the following relationships between the variance and mean is expected?
Show Hint
- Variance \(>) mean
- Variance \(<) mean
- Variance = mean
- Variance is independent of the mean
The Correct Option is C
Solution and Explanation
Step 1: Understanding the distribution of trees.
For randomly distributed trees, the number of individuals per plot follows a Poisson distribution. In a Poisson distribution, the variance is equal to the mean. This implies that for random distribution, the variance and mean of the number of individuals per plot will be equal.
Step 2: Interpreting the options.
- (A) Variance \(>) mean: This would occur in situations of overdispersion, but it is not expected in a random distribution.
- (C) Variance = mean: This is the characteristic relationship for a Poisson distribution, where the variance equals the mean.
- (D) Variance is independent of the mean: This would not apply in this case because in a Poisson distribution, the variance is directly tied to the mean.
Step 3: Conclusion.
The correct answer is (C) Variance = mean, as expected for a random distribution of trees in the plots.
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