The gypsy moth is a serious threat to oak and aspen trees. A state agriculture d

Question

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 months. The distribution of moth counts is discrete and strongly skewed, with standard deviation 0.9. What is the mean (plus-minus 0.1) of the average number of moths x bar?in 30 traps? And the standard deviation? (plus-minus 0.001) Use the central limit theorem to find the probability (plus=-minus 0.01) that the average number of moths in 30 traps is greater than 0.5:

Explanation / Answer

=0.5, =0.7, n=50

According to the central limit theorem, the mean value of a sample with a large enough sample size is normally distributed.
If a sample of size n is drawn from a population with mean and standard deviation , then the sample mean Xbar is normally distributed with mean and standard deviation /n.

Any normally distributed variable X with mean and standard deviation ,
X ~ N( , ² ), can be transformed into standard normal distribution by
Z = ( X - ) / , where Z ~ N( 0, 1) ==>
The standard normal variable Z has =0 and =1.

There are tables of the normal distribution, where you can read the value
of the standard normal cumulative probability F(z) or (z).

(a) Here, according to the central limit theorem, the average number of moths in 50 traps is normally distributed with
mean(Xbar)==0.5 and
standard deviation==/n=0.7/50=0.098995

(b) z=(0.6-)/=(0.6-0.5)/0.098995=1.01015
P(Xbar>0.6)=P(z>1.01015)=1-F(1.01015)=
1-0.84379=0.15621

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