Question

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.

Answer 1

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

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(
`\mu=5.58,\sigma^(2) =0.04^(2)`)

The z score probability distribution is given by;

Z =
`(X-\mu)/(\sigma)` ~ N(0,1)

where,
`\mu` = mean diameter = 5.58 millimeter

`\sigma` = standard deviation = 0.04 millimeter

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

• Firstly, Probability that the diameter separate the top 5% is given by;

P(X > x) = 0.05

P(
`(X-\mu)/(\sigma)` >
`(x-5.58)/(0.04)` ) = 0.05

P(Z >
`(x-5.58)/(0.04)` ) = 0.05

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

`(x-5.58)/(0.04)` = 1.6449

`{x-5.58} = 1.6449 * {0.04}`

`x` = 5.58 + 0.065796 = 5.65

• Secondly, Probability that the diameter separate the bottom 5% is given by;

P(X < x) = 0.05

P(
`(X-\mu)/(\sigma)` <
`(x-5.58)/(0.04)` ) = 0.05

P(Z <
`(x-5.58)/(0.04)` ) = 0.05

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

`(x-5.58)/(0.04)` = -1.6449

`{x-5.58} = -1.6449 * {0.04}`

`x` = 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.