乡亲们
让我们来看看常春藤大学的一名研究生......
约翰-福斯特感到非常幸运。他不仅以最优异的成绩拿到了本科学位，现在还是常春藤大学的研究生，师从著名的帕蒂-科尔曼教授。正当他沉浸在这些愉快的想法中时，科尔曼教授走到了他的桌前，他正在做高级统计学的家庭作业。
“Hey, John, I have a little assignment for you. Mike Madigan, CEO of ACME, has a vendor that is guaranteeing zero defects in lots of diodes that ACME buys, yet when ACME gets the lots they find a defect rate of around 1% or more. Can you contact ACME’s quality engineer, Frank Ianonne, to see how you can help?" Patty asked. “We covered this topic in the intro stats class you took last term,” she finished.
"当然，乐意效劳。"约翰答道。
"帕蒂感激地说："谢谢，我第一次参加SMTA 泛太平洋会议，有很多事情要做。
"哇，"约翰想，"压力山大啊。"
John contacted Frank and learned that the vendor’s sales engineer, Mike Gladstone, said that they sample 20 diodes from each 10,000-part lot. If they find no defects in in the sample of 20, they claim they can say that there are 0 defects with 95% confidence, since 19 out of 20 is 95% and they found no defects.
"呀，"约翰想，"这不可能是对的。"
他思前想后，特别是在看了科尔曼教授提到的那堂课的笔记后，终于想出了一个他确信无疑的答案。他联系了弗兰克，他们安排了一次与迈克的通话来讨论这个问题。
On the Zoom call after introductions, Frank asked Mike how they determine that a lot has zero defects.
"迈克说："我很高兴有机会向你们解释这个问题。
约翰觉得他的语气似乎很傲慢。
迈克继续说："那么，你会同意 20 分中有 19 分是 95%吧？
"是的，"弗兰克和约翰回答道。
“So, if we don’t get no defects in 20 samples, we got zero defects in the lot with 95% confidence. If we had one defect in the 20 samples, we couldn’t claim to have no zero defects in the lot,” Mike said.
“Mike, look at the image I took of one defect (a red bead) out of 2000 beads." (See Figure 1.) "If I selected 20 beads on the leftside of the container, how would I know that the defect rate is 0.0005 (1 in 2000)?” asked John.
Figure 1. The red bead is one “defect” out of 2000.
长时间的沉默。
"弗兰克问道："迈克，你的答案是什么？
还是没有回音。
“The answer is that the only way you can assure zero defects is to evaluateall of the samples,” said John.
"你只是在用那张照片混淆视听，"迈克吐出一句。
"弗兰克说："我觉得很清楚。
"你们常春藤联盟的人都一样。你们用胡言乱语混淆视听，而任何傻瓜都能看出我是对的。"迈克大叫道。
随后，一些脏话接踵而至，弗兰克切断了迈克的 Zoom 连接。
"我明白你的意思，约翰，"弗兰克说。"但是，你能给我一些数学上的支持吗？"
"当然，"约翰回答道。
“Let’s consider a case where the defect rate is not zero, but quite low, say 1 in 10,000 in a very large population. When we select the first sample, the likelihood of it being good is 0.9999 (10,000-1)/10000). What is the likelihood that the second one will be good?” John asked.
"啊，让我想想......0.9999，对吗？弗兰克回答道。
"但这两件事发生的可能性有多大？"约翰问道。
"等等，我记得几年前我上过统计课，它是 0.9999 x 0.9999，"弗兰克得意地说。
"连续三次都是好的可能性有多大？"约翰又问。
"0.99993，"弗兰克自信地回答。
"那么，假设我们取样很多次，我们称之为 n 次，0.9999n = 0.05。这说明了什么？"约翰问道。
"嗯，.........，"弗兰克回答道。
“Well, how likely is this to happen if the defect rate is 1 in 10,000?" John asked.
"等等，我明白了，这只会发生在0.05%或5%的情况下"。弗兰克兴奋地回答道。
“So, let’s say we didn’t know the defect rate, what could we say if we sampled n and got no defects?” queried John.
弗兰克被难住了。
"这样吧，你考虑一下，我们明天再来。现在已经快 6 点了。对了，看看你能不能算出 n 是多少。让我们在上午 10 点进行放大。"约翰提议道。
时间过得很快，约翰和弗兰克又开始 "Zooming "了。
"约翰，你差点要了我的命，我失眠了，但在查看了我的统计手册和做了一些 Youtubing 之后，我想我知道了。
“Well, if we didn’t know the defect rate and wanted to see if it was at least as good as 1 in 10,000 and we sampled n such that 0.9999n = 0.05, we could say with 0.95 (1 – 0.05) confidence that the defect rate was 1 in 10,000 or less,” Frank said triumphantly.
"正是如此，"约翰感叹道。
"但 n 是什么？"约翰问道。
"这就是我卡住的地方。我们有方程 0.9999n= 0.05，但我无法求出 n，"弗兰克沮丧地说。
"提示：对数，"约翰回答。
"就是这样，我知道了，"弗兰克热情地说。
弗兰克用计算器算了几分钟，得出了图 2 所示的解决方案。
Figure 2. The Defects Calculation
“So, to show with 95% confidence that the defect rate is 1 in 10,000 or less, we would have to sample almost 30,000 components and find no defects,” Frank exclaimed.
“By looking at the equation, you can see if the defect rate was zero, 0.9999 would be replaced by 1 and the log of 1 is 0 so you would need an infinite sample,” said John.
“So, the only way to show 0 defects is to sample all of the components,” Frank said.
"对！"约翰答道。
干杯
罗恩博士
