Question

The diameters of bolts produced in a machine shop are normally distributed with a mean of 5.67 millimeters and a standard deviation of 0.03 millimeters. Find the two diameters that separate the top 9% and the bottom 9%. These diameters could serve as limits used to identify which bolts should be rejected. Round your answer to the nearest hundredth, if necessary.

Answer 1

Solution:

Given:The diameters of bolts produced in a machine shop are normally distributed with a mean of 5.67 millimeters and a standard deviation of 0.03 millimeters.

That is: mm

mm

We have to find  two diameters that separate the top 9% and the bottom 9%.

That is:

P( X < x1 ) =0.09 and P( X > x2) =0.09

Thus first find z value such that: P( Z< z) = 0.09

Look in z  table for Area = 0.0900 or its closest area and find corresponding z value.

area 0.0901 is closest to 0.0900 and it corresponds to -1.3 and 0.04

thus z = -1.34

That is:

P( Z < -1.34) =0.09  

Since Normal distribution is symmetric,

P( Z < -z ) = P( Z > z)

Thus for upper 0.09 area z value = 1.34

Now use following formula to find x value:

and

Thus  the two diameters that separate the top 9% and the bottom 9% are 5.63 mm and 5.71 mm respectively.