A machine manufactures tires with a tread thickness that is normally distributed with mean 10 millimeters (mm) and standard deviation 2 mm. The tire has a 50,000- mile warranty. In order to last for...

A machine manufactures tires with a tread thickness that is normally distributed with mean 10 millimeters (mm) and standard deviation 2 mm. The tire has a 50,000- mile warranty. In order to last for 50,000 miles the tread thickness must be at least 7.9 mm. If the thickness of tread is measured to be less than 7.9 mm, then the tire is sold as an alternative brand with a warranty of less than 50,000 miles.

(a) Find the expected proportion of tires sold under the alternative brand.

(b) The demand for the alternative brand of tires is such that 30% of the total output should be sold under the alternative brand name. What should the critical thickness, originally 7.9 mm, be set at in order to meet the demand?