Question 1019893
 
Question:
The lengths of nails produced in a factory are normally distributed with a mean of 
4.95 centimeters and a standard deviation of 
0.05 centimeters.  Find the two lengths that separate the top 
5% and the bottom 
5%.  These lengths could serve as limits used to identify which nails should be rejected.  Round your answer to the nearest hundredth, if necessary.
 
Solution:
The top and bottom 5% can be obtained as a multiple of the standard deviation (σ) using the normal distribution curve, and the definition of Z=(X-μ)/σ.
 
From normal distribution tables, the 95% and 5% cutoffs are Z=±1.65.
Since Z=(X-μ)/σ, we solve for X in terms of Z
X=±1.65σ+μ
=5.95±1.65*0.05=(4.868,5.032)
are the limits to the 5-95% bracket of lengths.