The diameters of bolts produced in a machine shop are normally distributed with a mean of 5.61 millimeters and a standard deviation of 0.04 millimeters. Find the two diameters that separate the top 7% and the bottom 7%. These diameters could serve as limits used to identify which bolts should be rejected. Round your answer to the nearest hundredth, if necessary.

Answer:The two diameters that separate the top 7% and the bottom 7% are 5.6692 mm and 5.5508 mm respectively .Step-by-step explanation:Given :Mean = 5.61 millimeters            Standard deviation = 0.04 millimeters.To Find : Find the two diameters that separate the top 7% and the bottom 7%Solution :Case 1 We need to find [tex]x_1[/tex] such that [tex]P(X>x_1)= 7\% = 0.07[/tex] [tex]P(X\leq x_1)=1-0.07 = 0.93[/tex]Refer the z table for z value So, z corresponding to p = 0.97 is 1.48Formula : [tex]z=\frac{x_1-\mu}{\sigma}[/tex][tex]1.48=\frac{x_1-5.61}{0.04}[/tex][tex]1.48 \times 0.04=x_1-5.61[/tex][tex]0.0592=x_1-5.61[/tex][tex]0.0592+5.61=x_1[/tex][tex]5.6692=x_1[/tex]So, the diameter that separate the top 7% is [tex]5.6692=x_1[/tex]Case 2)We need to find [tex]x_2[/tex] such that [tex]P(X<x_2)= 7\% = 0.07[/tex]Refer the z table for z value So, z corresponding to p = 0.07 is -1.48Formula : [tex]z=\frac{x_2-\mu}{\sigma}[/tex][tex]-1.48=\frac{x_2-5.61}{0.04}[/tex][tex]1.48 \times 0.04=x_2-5.61[/tex][tex]-0.0592=x_2-5.61[/tex][tex]-0.0592+5.61=x_2[/tex][tex]5.5508=x_2[/tex]So, the diameter that separate the bottom 7% is [tex]5.5508=x_2[/tex]Hence  the two diameters that separate the top 7% and the bottom 7% are 5.6692 mm and 5.5508 mm respectively .