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

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 $x_1$ 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 : $z=\frac{x_1-\mu}{\sigma}$

$1.48=\frac{x_1-5.61}{0.04}$

$1.48 \times 0.04=x_1-5.61$

$0.0592=x_1-5.61$

$0.0592+5.61=x_1$

$5.6692=x_1$

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 : $z=\frac{x_2-\mu}{\sigma}$

$-1.48=\frac{x_2-5.61}{0.04}$

$1.48 \times 0.04=x_2-5.61$

$-0.0592=x_2-5.61$

$-0.0592+5.61=x_2$

$5.5508=x_2$

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 .