ShinKai Optical is an optical accessories manufacturing company in Shenzhen, China, that produces components for major fashion spectacle manufacturers in Europe and the United States.

These corporations outsource the manufacture of some minor components to China to contain costs. They then import the parts and assemble the glasses in their headquarters.

ShinKai prides itself on manufacturing the components to the exact design and specifications provided. However, recently, quality control from one of the corporations in Germany has found that the actual size of a frame is not the required standard—between 0.99 and 1.01 centimeters—and thus does not correspond to the industrial design as prepared by the European customer.

The manager of the plant has claimed that this case was a bad accident and that, normally, the glasses produced conform to all standards imposed by the corporation’s headquarters in Europe.

To check his assertions, a new quality check has been carried out.

A random sample of 12 spectacle frames was taken for analysis, which resulted in the sample being normally distributed with a mean of 1 centimeter and a standard deviation of 0.1 centimeters.

a. Determine the standard error for the sample mean.

b. Evaluate the claim of the factory manager that it was a one-time accident by finding the actual probability of the sample mean being between 0.99 and 1.01 centimeters. ( LO,1)