The stiffness of a certain type of steel beam used in building construction has mean 30 kN/mm and standard deviation 2 kN/mm.
a. Is it possible to compute the probability that the stiffness of a randomly chosen beam is greater than 32 kN/mm? If so, compute the probability. If not, explain why not.
b. In a sample of 100 beams, is it possible to compute the probability that the sample mean stiffness of the beams is greater than 30.2 kN/mm? If so, compute the probability. If not, explain why not.
(a) No, we do not know the distribution of the population of beam stiffnesses. In particular, we do not know whether
the population is normal.
(b) Yes, the mean of a large sample is normally distributed, so the Central Limit Theorem can be used.
Let 
X be the sample mean stiffness. The is approximately normally distributed with mean 
and standard deviation 
.
The z-score of 30.2 is (30.2 ? 30) / 0.2 = 1.00.
The area to the right of z = 1.00 is 1 ? 0.8413 = 0.1587.
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