The diameters of bolts produced in a machine shop are normally distributed with a mean of 5.65 millimeters and a standard deviation of 0.07 millimeters. Find the two diameters that separate the top 4% and the bottom 4%. These diameters could serve as limits used to identify which bolts should be rejected. Round your answer to the nearest hundredth, if necessary.

Two diameters that separate the top 4% and the bottom 4% are 5.77 and 5.53 respectively.

Step-by-step explanation:

We are given that the diameters of bolts produced in a machine shop are normally distributed with a mean of 5.65 millimeters and a standard deviation of 0.07 millimeters.

Let X = diameters of bolts produced in a machine shop

So, X ~ N()

The z score probability distribution is given by;

         Z = ~ N(0,1)

where, = population mean

             = standard deviation

Now, we have to find the two diameters that separate the top 4% and the bottom 4%.

        P(X > x) = 0.04

        P( > ) = 0.04

        P(Z > ) = 0.04

So, the critical value of x in z table which separate the top 4% is given as 1.7507, which means;

                   = 1.7507

                          = 5.65 + 0.122549 = 5.77

        P(X < x) = 0.04

        P( < ) = 0.04

        P(Z < ) = 0.04

So, the critical value of x in z table which separate the bottom 4% is given as -1.7507, which means;

                   = -1.7507

                          = 5.65 - 0.122549 = 5.53

Therefore, the two diameters that separate the top 4% and the bottom 4% are 5.77 and 5.53 respectively.