Reference no: EM131852868
Owing to variations in raw materials, preparation, etc., the quality Y of concrete varies from batch to batch. For a batch of concrete of given quality (Y = y), a specimen taken from the batch and tested by a standard procedure (itself subject to variation) will indicate "O.K." with probability, some increasing function of the quality. Thus this probability P varies from batch to batch. Assume that P has been found to have a quadratic distribution for a particular set of production and testing conditions.
fp (p) = 3p2 0 ≤ p ≤ 1
(a) What is the distribution of the number N of O.K. specimens in a sample of three (independent) specimens, if the quality is exactly equal to the desired quality y0 (with its associated probability p0 )?
(b) Express the joint distribution of N and P.
(c) What is the distribution of the number of O.K. specimens in an arbitrary batch?
(d) What is the probability that the quality of a particular batch is less than the desired quality given that two out of three specimens were found to be O.K.?
(e) A desirable property of a quality-control plan is that it not "indicate" good quality when quality is in fact low [e.g., part (d) above]. State and demonstrate, qualitatively, ways in which this probability can be reduced, using this example. Consider both the reliability of the testing procedure and the number of specimens per sample.