Question

Consider a manufacturing process called as turning (a type of machining process) that is used to manufacture cylindrical metal samples with a nominal diameter of 10.00 mm. Past data from the manufacturer shows a variation in the diameter of samples that is given by a normal distribution with a standard deviation of 0.50 mm. If the specified tolerance on the diameter is 0.75 mm, roughly what percentage of samples manufactured using this process satisfy the tolerance specification?

Answer 1

Answer: 86.64%

Explanation:

Let x be a random variable that represents the diameter of metal samples.

Given : Population mean :
`\mu=10`

Standard deviation:
`s=0.50`

Specified tolerance on the diameter is 0.75 mm.

i.e. range of diameter = 10-0.75< x <10+0.75 = 9.25< x< 10.75

Formula to find the z-score corresponds to x:
`z=(x-\mu)/(s)`

At x= 0.75,
`z=(9.25-10)/(0.50)=-1.5`

`z=(9.25-10)/(0.50)=1.5`

Using standard normal table for z-value,

P-value :
`p(-1.5<x<1.5)=1-2(P(z>1.5))\\\\=1-2(0.0668072)=0.8663856\approx0.8664=86.64\%`

∴ Percentage of samples manufactured using this process satisfy the tolerance specification = 86.64%