(15.12) The gypsy moth is a serious threat to oak and aspen trees. A state agric

Question

(15.12) 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 0.5, but some traps have several moths. The distribution of moth counts is discrete and strongly skewed, with standard deviation 0.9. What is the mean (±0.1) of the average number of moths x in 30 traps? And the standard deviation? (±0.001) Use the central limit theorem to find the probability (±0.01) that the average number of moths in 30 traps is greater than 0.8:

Explanation / Answer

What is the mean (±0.1) of the average number of moths x in 30 traps?

By central limit theorem, the mean remains the same,

u(X) = u = 0.5 [ANSWER]

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And the standard deviation? (±0.001)

By central limit theorem, the standard deviation is reduced,

sigma(X) = sigma/sqrt(n) = 0.9/sqrt(30) = 0.164316767. [ANSWER]

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Use the central limit theorem to find the probability (±0.01) that the average number of moths in 30 traps is greater than 0.8:

We first get the z score for the critical value. As z = (x - u) sqrt(n) / s, then as          
          
x = critical value =    0.8      
u = mean =    0.5      
n = sample size =    30      
s = standard deviation =    0.9      
          
Thus,          
          
z = (x - u) * sqrt(n) / s =    1.825741858      
          
Thus, using a table/technology, the right tailed area of this is          
          
P(z >   1.825741858   ) =    0.033944577 = 0.03 [ANSWER]

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