from a normally distributed infinite number of iron bars lengths with a mean and standard deviation as 4 ft. and 0.6 ft. respectively, a sample of 100 bars is taken. if the sample mean is 4.2 ft., can the sample be called a truely random sample?

Answer 1

How was the sample taken? If the observations are taken on a chance basis, then it would be random.

Or are you asking if the sample is within certain range of possible sample means?

Z = (score-mean)/SEm

SEm = SD/√n

Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/probability related to the Z score.