Suppose that a project manager tested the strength of one tile out of a batch of 3,000 to be used on a building. This one sample measurement was compared with the design specification and, in this case, the sampled tile's strength exceeded that of the specification. On this basis, the project manager accepted the tile shipment. If the sampled tile was defective (with a strength less than the specification), the project manager would have rejected the lot. Required:
a. What is the probability that ninety percent of the tiles are substandard, even though the project manager's sample gave a satisfactory result?
b. Sketch out the operating characteristic curve for this sampling plan as a function of the actual fraction of defective tiles.