The lifetime of an insulating The lifetime of an insulating

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.

Solution

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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