The diameters of bolts produced in a machine shop are normally distributed with a mean of 5.585.58 millimeters and a standard deviation of 0.040.04 millimeters. Find the two diameters that separate the top 5%5% and the bottom 5%5%. These diameters could serve as limits used to identify which bolts should be rejected. Round your answer to the nearest hundredth, if necessary.

The two diameters that separate the top 5% and the bottom 5% are 5.51 and 5.65 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 5.58 millimeters and a standard deviation of 0.04 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, = mean diameter = 5.58 millimeter

             = standard deviation = 0.04 millimeter

Now, we have to find the two diameters that separate the top 5% and the bottom 5%.

        P(X > x) = 0.05

        P( > ) = 0.05

        P(Z > ) = 0.05

So, the critical value of x in z table which separate the top 5% is given as 1.6449, which means;

                       = 1.6449

                               = 5.58 + 0.065796 = 5.65

        P(X < x) = 0.05

        P( < ) = 0.05

        P(Z < ) = 0.05

So, the critical value of x in z table which separate the bottom 5% is given as -1.6449, which means;

                       = -1.6449

                               = 5.58 - 0.065796 = 5.51

Therefore, the two diameters that separate the top 5% and the bottom 5% are 5.51 and 5.65 respectively.

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