Question

The diameters of bolts produced in a machine shop are normally distributed with a mean of 5.65 millimeters and a standard deviation of 0.07 millimeters. Find the two diameters that separate the top 4% and the bottom 4%. 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

To find the diameters that separate the top and bottom 4% of normally distributed bolts, calculate the z-scores for these percentiles and convert them to diameters using the given mean (5.65 mm) and standard deviation (0.07 mm). The diameters are 5.52 mm for the bottom 4% and 5.78 mm for the top 4%.

Step-by-step explanation:

To find the two diameters that separate the top 4% and the bottom 4% of bolts in a normally distributed set, we need to find the respective z-scores and then convert them to diameters using the mean and standard deviation.

Calculation for the bottom 4%

The z-score corresponding to the bottom 4% can be found using a standard normal distribution table or a z-score calculator. Based on the standard normal distribution, the z-score for the bottom 4% is approximately -1.750. Using the mean μ = 5.65 mm and standard deviation σ = 0.07 mm, we convert the z-score to the diameter:
X = μ + zσ = 5.65 + (-1.750)(0.07) = 5.52 mm (rounded to the nearest hundredth).

Calculation for the top 4%

Similarly, the z-score for the top 4% is approximately +1.750. Converting this z-score to the diameter:
X = μ + zσ = 5.65 + (1.750)(0.07) = 5.78 mm (rounded to the nearest hundredth).

Therefore, the bolts with diameters less than 5.52 mm or greater than 5.78 mm should be rejected.

Answer 2

Two diameters that separate the top 4% and the bottom 4% are 5.77 and 5.53 respectively.

Step-by-step explanation:

We are given that the diameters of bolts produced in a machine shop are normally distributed with a mean of 5.65 millimeters and a standard deviation of 0.07 millimeters.

Let X = diameters of bolts produced in a machine shop

So, X ~ N(
`\mu=5.65,\sigma^(2) = 0.07^(2)`)

The z score probability distribution is given by;

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

where,
`\mu` = population mean

`\sigma` = standard deviation

Now, we have to find the two diameters that separate the top 4% and the bottom 4%.

• Firstly, Probability that the diameter separate the top 4% is given by;

P(X > x) = 0.04

P(
`(X-\mu)/(\sigma)` >
`(x-5.65)/(0.07)` ) = 0.04

P(Z >
`(x-5.65)/(0.07)` ) = 0.04

So, the critical value of x in z table which separate the top 4% is given as 1.7507, which means;

`(x-5.65)/(0.07)` = 1.7507

`x-5.65 = 0.07 * 1.7507`

`x` = 5.65 + 0.122549 = 5.77

• Secondly, Probability that the diameter separate the bottom 4% is given by;

P(X < x) = 0.04

P(
`(X-\mu)/(\sigma)` <
`(x-5.65)/(0.07)` ) = 0.04

P(Z <
`(x-5.65)/(0.07)` ) = 0.04

So, the critical value of x in z table which separate the bottom 4% is given as -1.7507, which means;

`(x-5.65)/(0.07)` = -1.7507

`x-5.65 = 0.07 * (-1.7507)`

`x` = 5.65 - 0.122549 = 5.53

Therefore, the two diameters that separate the top 4% and the bottom 4% are 5.77 and 5.53 respectively.