The lengths of nails produced in a factory are normally distributed with a mean of 5.13 centimeters and a standard deviation of 0.04 centimeters. find the two lengths that separate the top 8% and the bottom 8%. these lengths could serve as limits used to identify which nails should be rejected. round your answer to the nearest hundredth, if necessary.

The standard normal table starts at .5 for z=0, meaning its giving us the integral of the standard gaussian from negative infinity to some positive z.  We look for a table entry of 1-.08=.92 and find that at z=1.41.

We knew it was between one and two because the 68-95-99.7 rule tells us one sigma gives a probability of (100-68)/2=16% and two sigma 2.5%, and 8% is in between those.

So our lengths at the ends of the range are 

5.13 - 1.41(0.04) = 5.0736

5.13  + 1.41(0.04) = 5.1864

Rounding, 5.07 to 5.19