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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?

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JeanaShupp

Answer: 86.64%

Step-by-step explanation:

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

Given : Population mean : [tex]\mu=10[/tex]

Standard deviation: [tex]s=0.50[/tex]

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: [tex]z=\dfrac{x-\mu}{s}[/tex]

At x= 0.75,  [tex]z=\dfrac{9.25-10}{0.50}=-1.5[/tex]

[tex]z=\dfrac{9.25-10}{0.50}=1.5[/tex]

Using standard normal table for z-value,

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

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

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