1. A forester studying the effects of fertilization on certain pine forests in t
ID: 3028108 • Letter: 1
Question
1.
A forester studying the effects of fertilization on certain pine forests in the Southeast is interested in estimating the average basal area of pine trees. In studying basal areas of similar trees for many years, he has discovered these measurements (in square inches) to be normally distributed with standard deviation approximately 4 square inches. Say the forester samples n=9 trees. Suppose the forester would like the sample mean to be within 1 square inches of the population mean, with probability .90.
A. How many trees must he measure in order to ensure this degree of accuracy?
2.
has an F distribution with v1 and v2 numerator and denominator degrees of freedom, respectively. Use the preceeding structure of F, the independence of W1 and W2 to show
A.) E(F)=v2/(v2-2) if v2 >2
PlglExplanation / Answer
1) A) B)
Probability that the sample mean will be withing 1 square inches of the population mean:
Let Mu be the population mean and xbar the sample mean
P ( Mu-1 < xbar < Mu+1)
P( ( Mu-1-Mu)/(sd/sqrt(n)) < (xbar - mu) /(sd/sqrt(n)) < (Mu+1 -Mu)/sd/sqrt(n))
sd/sqrt(n) = 4/sqrt(9) = 4/3
P( -1/(4/3) < z < 1 /(4/3))
= P( -3/4 < z < 3/4)
= P( - 0.75 < z < 0.75) ----------------------------- This is the answer of Part (A)
B) g1 and g2 are -0.75 and 0.75
2)
A) The moment generating function of the F distribution is
E(X^n) = [ ( (v1 + 2) / 2) ( (v2 - 2) / 2) ] / [ (v1/2) (v2/2) ] * (v2/v1)^n
where (n) is the gamma function. Remember that (n) = (n - 1)! and (a + 1) = a (a)
The expectation is found by evaluating the moment generating function with n = 1
[ (v1/2 + 1) (v2/2 - 1) ] / [ (v1/2) (v2/2) ] * (v2/v1)
[(v1/2)(v1/2) (1/ (v2/2 - 1))(v2/2) ] / [ (v1/2) (v2/2) ] * (v2/v1)
[(v1/2) (1/ (v2 - 2)) ] * (v2/v1)
v1 / 2 * 2 / (v2 - 2) * v2 / v1
= v2 / (v2 - 2)
B) the variance of the F distribution is:
= 2 * ( v2/ (v2-2) ) ^ 2 * (v1 + v2 - 2) / (v1 ( v2 + 4) ) for v2 > 4
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