A manufacturer has contracted to supply ball bearings. Product analysis reveals that the diameters are normally distributed with a mean 25.1 mm and a standard deviation of 0.2 mm. The largest 7% of the diameters and the smallest 13% of the diameters are unacceptable. Find the limits for the diameters of the acceptable ball bearings.
We are given a normally distributed population with `mu=25.1` and `sigma=.2` . We are asked to find the diameters that lie between the bottom 13% and the top 7%.
(1) We find the z-score that has 13% of the area under the standard normal curve to the left of it. Consulting a standard normal table or technology we find `z~~-1.48` .
(2) We find the z-score such that 7% of the area under the standard normal curve is to the right of it. This is equivalent to finding 93% to the left. From a table or technology we get `z~~1.88`
(3) From `z=(x-mu)/sigma` we see that `x=zsigma+mu` .
The x-value associated with z=-1.48 is x=-1.48(.2)+25.1=24.8 and the x-value associated with a z-score of 1.88 is x=1.88(.2)+25.1=25.48
Thus the permissable diameters lie between 24.8mm and 25.5mm