The lifetime of an insulating The lifetime of an insulating material tested (acc

Question

The lifetime of an insulating

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 > 65.20). For the following scenarios, describe what the mean p represents, and set up H_0 and H_a related to mu (i.e. What hypotheses would you test to assess the specification/claim/belief?) Do not perform the hypothesis tests. A random sample of 25 pieces of acetate fiber has a sample mean absorbency of 12% with a sample standard deviation of 1.25%. Is there strong evidence that this fiber has a true mean absorbency of less than 20%? A manufacturer of a synthetic fishing line claims that its product has a mean breaking strength of at least 8 kilograms. A random sample of n=9 fishing line specimens yields a sample average strength of 7.5 kilograms with sample standard deviation of 0.65 sec.

Explanation / Answer

2.

a)

Here, the notation of a normal distrbution is N(mean, variance).

Note that

f(Xbar) = N~(u, sigma^2/n)

Then, the distribution of Xbar is

f(Xbar) = N~(63, 8^2/50)

f(Xbar) = N~(63, 1.28) [ANSWER]

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b)

We first get the z score for the critical value. As z = (x - u) / s, then as          
          
x = critical value =    65.2      
u = mean =    63      
          
s = standard deviation =    1.13137085      
          
Thus,          
          
z = (x - u) / s =    1.944543648      
          
Thus, using a table/technology, the right tailed area of this is          
          
P(z >   1.944543648   ) =    0.025914964 [ANSWER]

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