Question

(Gloves, 60 pts). Tertemiz Inc. produces powdered latex gloves for daily use and sells them in boxes of size K = 100. In the production line, each glove is produced independently with a fixed probability s = 0.002 of being defective. For the time being, let us consider a generic glove box. Let Z be the number of defective gloves in the box.

e. (10) Calculate the probability that the lot is accepted. With the new coronavirus outbreak, the glove boxes of Tertemiz Inc. have been in high demand recently. Therefore, the boxes are shipped to the central warehouses of supermarkets in lots of size N2 = 1000 to be further distributed to the branches. Now, let us consider a fixed lot of size N2. The company keeps using the same acceptance sampling procedure with sample size n = 5 and acceptance number c = 1 as describedbefore. Asbefore,wedenoteby Y thetotalnumberofdefectiveboxesinthe lot and by X the total number of defective boxes in the sample.

f. (10)Let y ∈{0,...,N2}. Ifitisknownthatthereare Y = y defectiveboxesinthe lot, then calculate the probability that the lot is accepted. Hint: Due to the high demand assumption, the sample size is now 0.5% of the lot size. Hence, it is safe to assume that the conditional distribution of X given Y = y is a binomial distribution. g. (10) Calculate the probability that the lot is accepted. Hint: Express this probability as a summation. This summation is not analytically tractable. Evaluateitnumericallyonacomputerbycalculatingeachtermseparately. Submit the value of each term as part of your solution (e.g., as a table/spreadsheet within your pdf).

Answer 1