A die casting process is being used to fabricate a cast housing for a major clie

Question

A die casting process is being used to fabricate a cast housing for a major client. The part weight of these housings is a critical quality concern that must satisfy the customer-specified tolerance of 770 notequalto 25 grams. A random sample of 40 housings has been taken from the process that has been running in control. The table below displays the measured weights of the sampled housings. 775 773 778 774 775 775 772 780 776 772 774 778 778 781 772 776 774 774 775 775 784 781 781 773 774 777 775 780 776 774 778 775 777 780 772 773 773 777 774 775 For each housing that weighs 745 grams and is put into field use, the customer will incur a loss of $250. If a part is rejected at the plant because it is either too light or too heavy to meet the specification, it is scrapped at a cost of.$15. a)Estimate the probability that a housing produced by this process will weigh within the specified tolerance interval assuming that the part weights follow a normal distribution with mean and standard deviation estimated from the sample data. b)Find the expected total quality cost per part produced under the current operating conditions and that meet the customer's requirement. (Assume a normal distribution and quadratic loss function centered at the specific nominal.)

Explanation / Answer

a. Let the weight be x. Then we have, from the given sample, E(X) = 776.11 and Std dev(X) = 3.022. Thus the estimated probability is given as

P( 770-25 < X < 770+25 ) = P(745<X<795) = P(-10.3<Z<6.25) = P(Z<6.25) - P(Z<-10.3) = 1 - 0 = 1

b. Since P(X<745) = P(Z<-10.3) = 0, total expected quality cost = 0

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