Question 1192349: The gypsy moth is a serious threat to oak and aspen trees. A state agriculture department places traps throughout the state to detect the moths. When traps are checked periodically, the mean number of moths trapped is only 0.4, but some traps have several moths. The distribution of moth counts is discrete and strongly skewed, with standard deviation 0.9.
A) What is the mean (±0.1) of the average number of moths x in 60 traps?
B) And the standard deviation? (±0.001)
2) Use the central limit theorem to find the probability (±0.01) that the average number of moths in 60 traps is greater than 0.6
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! **A) Mean and Standard Deviation of the Average Number of Moths**
* **Mean of the sample means (Central Limit Theorem):**
* The mean of the sample means is equal to the population mean:
* Mean of sample means = 0.4 moths
* **Standard deviation of the sample means (Standard Error):**
* Standard Error (SE) = σ / √n
* where:
* σ is the population standard deviation (0.9 moths)
* n is the sample size (60 traps)
* SE = 0.9 / √60
* SE ≈ 0.116 moths
**B) Probability of the average number of moths being greater than 0.6**
* **Standardize the value 0.6:**
* Z-score = (X - μ) / σ
* where:
* X is the value we're interested in (0.6 moths)
* μ is the mean of the sample means (0.4 moths)
* σ is the standard deviation of the sample means (0.116 moths)
* Z-score = (0.6 - 0.4) / 0.116
* Z-score ≈ 1.72
* **Calculate the probability using the standard normal distribution:**
* We need to find P(Z > 1.72)
* Using a standard normal distribution table or a calculator, we find that P(Z > 1.72) ≈ 0.0427
**Therefore:**
* **A) Mean of the average number of moths: 0.4 moths**
* **B) Standard deviation of the average number of moths: 0.116 moths**
* **Probability that the average number of moths in 60 traps is greater than 0.6: 0.04 (or 4%)**
This analysis shows that while the average number of moths per trap is low, the probability of finding an average of more than 0.6 moths in 60 traps is relatively small (about 4%). This suggests that the overall moth population in the area may not be excessively high.
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