Vendor quality control is an integral part of a total quality system. A soft drink bottling company requires its vendors to produce bottles with an internal pressure strength of at least 300 kg/cm2. A vendor claims that its bottles have a mean strength of 310 kg/cm2 with a standard deviation of 5 kg/cm2. As part of a vendor surveillance program, the bottling company samples 50 bottles from the production and finds the average strength to be 308.6 kg/cm2.

(a) What are the chances of getting that sample average that was observed, or even less, if the assertion by the vendor is correct? (b) If the standard deviation of the strength of the vendor's bottles is 8 kg/cm2, with the mean (as claimed) of 310 kg/cm2, what are the chances of seeing what the bottling company observed (or an even smaller sample average)?