Question

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.

Answer 1

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