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# Distribution of Sample Mean - Linkages

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Three shafts are made and assembled in a linkage. The length of each shaft, in centimeters, is distributed as follows:

Shaft 1: l_1 ~ N(75,0.09)
Shaft 2: l_2 ~ N(60,0.16)
Shaft 3: l_3 ~ N(25,0.25)

The length of the linkage is calculated by l = l_1 + l_2 + l_3. Assume the shafts' length are independent to each other:

a) What is the distribution of sample mean l?

b) How can we define the boundaries such that P(1 <= b_1 or 1 >= b_2) = 0.00270?

c) Assume that under abnormal production condition, the mean length of shaft l changes to 80 with the variance unchanged, and distribution parameters of the other two shafts are not changed. Suppose a sample of size n (n = 25) linkage is randomly selected from the abnormal production condition, what is the distribution of the new sample mean, Xbar?

d) Suppose the two boundaries, b_1 and b_2, are still used, find P(b_1 <= Xbar <= b_2)?