Question

CASE 14 SAM PLING DISTRIBUTORS "We have to think more about our acceptance sampling decisions," said Buddy Abbot, head of the production department of Sam Pling Distributors. Sam Pling Distributors is a large company providing over twenty percent of the tire nuts used by American automobile manufacturers Listening attentively was Louis Costello, Abbot's assistant, who was charged with the sampling procedures used to ensure that out-going shipments conformed to the proper specifications noted on the order form. The key quality indicator of tire nuts was its inner circumference. This circumference was standardized for most types of tires used on traditio automobiles, but differed for truck tires, and for other specialized tires used on certain sports cars, vans, and military vehicles. Also attending the meeting that Monday morning was Jacqui Benny, consulting statistician, and Jonathan Carson, a member of Sam Pling's Management Information Systems department, and present as a liaison to any computer help needed passenger Costello rifled through some papers in front of him, and summed up the issue under discussion as follows: "We have these large orders of tire nuts that Stanley Laurel, over at packing, readies for shipping out to our customers. Stanley, as part of our standard operating procedure, lets us know when a shipment is ready to be sampled. Our current acceptance sampling process is . simply to randomly choose 25 nuts out of a batch, measure the inner circumference of each, and the average value. If this average value, what the statistician labels "x-bar", is even reasonably close to the specified value of the circumference, we go ahead and ship the entire batch out to the customer. Otherwise, if it's not that close, we assume that we may have messed up, and perform a complete inspection of all the tire nuts. We subject each nut to measurement on the Hardy machine. Those nuts not within .05 inches of the specified value are culled out. Of course, using the Hardy machine is very expensive, but it's better than shipping a batch to the customer with a relatively large proportion of unusable nuts. If that happens, the customer can invoke the standard penalty clause on us, and that kills our profit on the entire batch!" Jacqui, who was new to the company, and had only recently begun to consider the issue under discussion, added to Louis' dialogue by noting that "historical inspection records show that one thing that has been constant is the variability in circumference from nut to nut. Regardless of the value of the average circumference, the circumference values have always been normally distributed with a standard deviation of.025 inches. Therefore, if the average circumference of the entire batch is exactly the specified value, a large majority of the nuts are "usable," (understood industry-wide to mean within .05 inches of the nominal required value.) [Q1]: What proportion are usable? The problem is that the manufacturing parameters are complex, and that sometimes the average circumference of a batch is off by .02 inches; then, numerous nuts are unusable [Q2]: What proportion are usable if the average is off by.02 inches? 56 The manufacturing process apparently produces average circumferences in increments of .02; i.e., whenever the batch average is off, it's off by .02 inches. In a rare instance, it may be off by .04 inches, though I believe that being off that far is so rare that we can safely ignore that possibility." At this point, Buddy stood up and said, "Look. The issue on the table is this: we can't control what we get from manufacturing. But, we can control two things, at least to some degree - (1) sending out a batch whose average is off specification, instead of subjecting the batch to complete inspection on the Hardy machine. It's expensive to use the Hardy machine, but it's cheaper to use it than to send out a batch whose average is not at spec; (2) on the other hand, we don't want to use the Hardy machine when we don't have to. That is, we want to avoid_subiecting a batch to the Hardy machine when its average is on spec in the first place. Louis, I want you to work with Jacqui to come up with a sampling plan that protects against these unnecessary expenses. Use Jonathan if you need to." Later that week, Jacqui met with Louis to report on her preliminary thoughts. She began her presentation, "We can implement a procedure that is independent of the specified value. For example, let's call the spec value, we can then set up the following scenario, knowing that whether the batch average is under or over spec by .02 doesn't matter, due to symmetry: H o + or - .02 We can now design a sampling plan that randomly chooses 25 nuts and uses a decision rule as follows: if the sample average, x-bar, is within.01 of Ho, send out the batch without subjecting the batch to the Hardy machine; if x-bar is not within.01 of Ho, do subject the batch to the Hardy machine. Then, if the batch average equals the specified value, the probability is small that we erroneously subject the batch to the Hardy machine. We call this kind of error a "type I error", and the probability of its occurring is traditionally denoted by a. Likewise, if the batch average is off by.02, the probability is low that we erroneously send the batch out without subjecting it to the Hardy machine. This error is usually called a "type error", and the probability of its occurring is traditionally denoted by B." [Q3: With this sampling plan, what is the value of a? of ß? Louis is pleased with Jacqui's analysis, but wonders about the α and β values that go along with her proposed acceptance sampling plan. He then asks her whether they could devise a sampling plan having different α and β values. She responds, "Yes, if you allow a sample size other than 25." Louis then asks Jacqui to devise a sampling plan having α β .01. [Q4]: If you were Jacqui, what sampling plan would you devise?. At the next Monday meeting, Buddy digested the plan having each error probability being.01. As he thought further, however, he realized that a type I error, submitting the batch to the Hardy machine when it is unnecessary, while costly, is not as costly as a type II error, sending out a batch without using the Hardy machine, when the batch average is off by .02. He then asked Jacqui if a and 8 had to be equal. Jacqui answered that they did not have to be equal Buddy then requested Jacqui to devise a sampling plan that allowed a to be.02, while yielding a 8 of .005. [Q5]: If you were Jacqui, what sampling plan would you now devise? Buddy also decided to think further about what the proper trade-off should be between the relative values of α and β. His instinct told him that there were two contributing aspects of this trade-off. One is the relative cost of making each type of error, the other is perhaps more subtle the relative chances that the batch is on spec or not that is, the performance of the manufacturing process. [Q6: Discuss the logic of each of these aspects with respect to its relevance to the α, β trade-off.

Answer 1

Answer:

In given problem there are possibilities of two types of error

Aspects with respect to its relevance to the $\alpha$ and $\beta$ trade-off

$\alpha$ is the probability of type I error and $\beta$ is the probability of type II error.

There are two contributing aspects of this trade-off. One is the relative cost of making each type of error like $\alpha$ and $\beta$ the other is the relative chances that the batch is on spec or not that is the performance of the manufacturing process.