Question

The diameters of bolts produced in a machine shop are normally distributed with a mean of 6.48 millimeters and a standard deviation of 0.06 millimeters. Find the two diameters that separate the top 3% and the bottom 3% . 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

Explanation:

To find the two diameters that separate the top 3% and the bottom 3% of the bolts produced in the machine shop, we need to find the lower and upper bounds of the interval that contains the middle 94% of the data. This can be done using the standard normal distribution and the z-score.

First, we'll find the z-score that corresponds to the lower bound. To do this, we'll use the formula:

z = (x - μ) / σ

where x is the lower bound, μ is the mean of the data (6.48 mm), and σ is the standard deviation of the data (0.06 mm). We want to find the value of x such that the area to the left of x is equal to 3%:

z = (x - 6.48) / 0.06

z = -2.33

Next, we'll use the z-score to find the value of x (the lower bound). We'll use the standard normal distribution table to look up the value of -2.33 and find that it corresponds to a cumulative probability of 0.01. So, the lower bound is such that 1% of the data is less than or equal to this value. To find the value of x, we'll use the formula:

x = μ + zσ

x = 6.48 + (-2.33) * 0.06

x = 6.32

So, the lower bound of the interval that contains the middle 94% of the data is 6.32 millimeters. This is the diameter that separates the bottom 3% of the bolts from the rest.

Next, we'll find the upper bound in a similar way. To do this, we'll use the formula:

z = (x - μ) / σ

where x is the upper bound, μ is the mean of the data (6.48 mm), and σ is the standard deviation of the data (0.06 mm). We want to find the value of x such that the area to the left of x is equal to 97%:

z = (x - 6.48) / 0.06

z = 2.33

Using the standard normal distribution table, we find that the value of 2.33 corresponds to a cumulative probability of 0.99. So, the upper bound is such that 99% of the data is less than or equal to this value. To find the value of x, we'll use the formula:

x = μ + zσ

x = 6.48 + 2.33 * 0.06

x = 6.64

So, the upper bound of the interval that contains the middle 94% of the data is 6.64 millimeters. This is the diameter that separates the top 3% of the bolts from the rest.

Therefore, the two diameters that separate the top 3% and the bottom 3% of the bolts produced in the machine shop are 6.32 millimeters and 6.64 millimeters, rounded to the nearest hundredth.