Question

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 1

The two diameters that separate the top 7% and the bottom 7% are 5.6692 mm and 5.5508 mm respectively .

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
`x_1` such that
`P(X>x_1)= 7\% = 0.07`

`P(X\leq x_1)=1-0.07 = 0.93`

Refer the z table for z value

So, z corresponding to p = 0.97 is 1.48

Formula :
`z=(x_1-\mu)/(\sigma)`

`1.48=(x_1-5.61)/(0.04)`

`1.48 * 0.04=x_1-5.61`

`0.0592=x_1-5.61`

`0.0592+5.61=x_1`

`5.6692=x_1`

So, the diameter that separate the top 7% is
`5.6692=x_1`

Case 2)

We need to find
`x_2` such that
`P(X<x_2)= 7\% = 0.07`

Refer the z table for z value

So, z corresponding to p = 0.07 is -1.48

Formula :
`z=(x_2-\mu)/(\sigma)`

`-1.48=(x_2-5.61)/(0.04)`

`1.48 * 0.04=x_2-5.61`

`-0.0592=x_2-5.61`

`-0.0592+5.61=x_2`

`5.5508=x_2`

So, the diameter that separate the bottom 7% is
`5.5508=x_2`

Hence the two diameters that separate the top 7% and the bottom 7% are 5.6692 mm and 5.5508 mm respectively .