Question

The diameters of bolts produced in a machine shop are normally distributed with a mean of 5.755.75 millimeters and a standard deviation of 0.070.07 millimeters. Find the two diameters that separate the top 6%6% and the bottom 6%6%. 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 D be the random variable denoting the diameter of this shop's bolts, so that D is normally distributed with µ = 5.75 and σ = 0.07. The top 6% and bottom 6% of bolts have diameters d₁ and d₂ such that

P(d₁ < D < d₂) = P(D < d₂) - P(D < d₁) = 0.94 - 0.06

i.e. d₂ is the 94th percentile and d₁ is the 6th percentile, for which

P(D < d₂) = 0.94

P(D < d₁) = 0.06

Convert D to a random variable Z following the standard normal distribution using

Z = (D - µ) / σ

Then

P(D < d₂) = P((D - 5.75) / 0.07 < (d₂ - 5.75) / 0.07)

0.94 = P(Z < (d₂ - 5.75) / 0.07)

→ (d₂ - 5.75) / 0.07 ≈ 1.55477

→ d₂ ≈ 5.86

P(D < d₁) = P((D - 5.75) / 0.07 < (d₁ - 5.75) / 0.07)

0.06 = P(Z < (d₁ - 5.75) / 0.07)

→ (d₁ - 5.75) / 0.07 ≈ -1.55477

→ d₁ ≈ 5.64

So bolts with a diameter between 5.64 mm and 5.86 mm are acceptable.