The lifetime of an insulating material tested accelerated te

The lifetime of an insulating material tested (accelerated testing) at 30kV is known to have a mean of 63 hours and standard deviation of 8 hours. A reliability engineer randomly selects n = 50 specimens and determines their lifetimes. What is the approximate distribution of X the sample average of the 50 lifetimes? (Use Central Limit Theorem.) Use the result of part (a) to approximate the probability P(X GE 65.20).

Solution

A)

The sampling distribution of X will have the same mean, but a variance of sigma^2/n.

Thus,

X ~ N(63, 8^2/50)

or

X ~ N(63, 1.28) [ANSWER, sampling mean 63, sampling variance of 1.28]

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B)

We first get the z score for the critical value. As z = (x - u) sqrt(n) / s, then as          
          
x = critical value =    65.2      
u = mean =    63      
n = sample size =    50      
s = standard deviation =    8      
          
Thus,          
          
z = (x - u) * sqrt(n) / s =    1.944543648      
          
Thus, using a table/technology, the right tailed area of this is          
          
P(z >   1.944543648   ) =    0.025914964 [ANSWER]