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%

Step-by-step 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: $z=\dfrac{x-\mu}{s}$

At x= 0.75,  $z=\dfrac{9.25-10}{0.50}=-1.5$

$z=\dfrac{9.25-10}{0.50}=1.5$

Using standard normal table for z-value,

P-value : $p(-1.5$

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