Question

A metallic split ring has a target internal diameter of 50 mm, but records show that the diameters are normally distributed

with mean 50 mm and standard deviation 0.07 mm.
An acceptable diameter is one within the range 49.93 mm to 50.07 mm. What proportion of the output is unacceptable?
Round your answer to four decimal places.

Answer 1

The proportion of unacceptable output is 93.64%.

Step-by-step explanation:

To find the proportion of unacceptable output, we need to find the proportion of diameters that fall outside the acceptable range of 49.93 mm to 50.07 mm.

We can do this by finding the cumulative probability of values less than 49.93 and greater than 50.07 using the normal distribution.

Using the mean of 50 mm and standard deviation of 0.07 mm, we can calculate the z-scores for the lower and upper limits.

The z-score for 49.93 is (49.93 - 50) / 0.07 = -1.857 and the z-score for 50.07 is (50.07 - 50) / 0.07 = 1.857.

Using a standard normal distribution table or a calculator, we can find the cumulative probability for z-scores less than -1.857 (which is 0.0318) and greater than 1.857 (which is 0.0318).

To find the proportion of unacceptable output, we subtract these probabilities from 1: 1 - 0.0318 - 0.0318 = 0.9364 or 93.64%.