Question

The diameters of bolts produced in a machine shop are normally distributed with a mean of 5.34 millimeters and a standard deviation of 0.03 millimeters. Find the two diameters that separate the top 8% and the bottom 8%. 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

Let X be a random variable such that

X: diameter of bolts produced in the machine shop.

Given that: X is normally distributed with mean, & standard deviation, .

Thus, it can be denoted as .

Let the two diameters that separate the bottom 8% and the top 8% be denoted by k₁ and k₂ respectively.

k₁ and k₂ are such that,

P(x < k₁) = 0.08 and P(x > k₂) = 0.08

or P(x < k₁) = 0.08 and P(x < k₂) = 1 - 0.08 = 0.92

Now let , such that z ~ N(0,1) and is known as standard normal variate.

Thus, the required probabilities can be written as:

, and

Using the normal distribution table, the values of z₁ and z₂ are obtained as:

z₁ = -1.4051 and z₂ = 1.4051

Replacing x with k₁ and k₂ & z with z₁ and z₂ in the equation , we have

.