# Formulate a decision criterion and then solve the decision problem for under given situation Suppose in a battery manufacturing plant 20 batteries are taken at random from each machine’s...

- Formulate a decision criterion and then solve the decision problem for under given situation

- Suppose in a battery manufacturing plant 20 batteries are taken at random from each machine’s production in a shift. The manufacturer would like to ensure that the chance of producing a defective battery should not exceed

0.01. If the chance is more than that, then the production should be stopped

and engineers should hunt for the causes of failure. Based on the sample of

size 20 the shift manager has to decide whether to stop the process or not. Of

course, there is a chance that even if the process proportion of defective is

0.01 or less, all 20 batteries in the sample may be defective.

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### 2 Answers

If the probability of the the defective battery is 0.01 or less in the manufactured lot, then the the probability that any battery is not defective should be 0.99 or more than 0.99. So the probabality that none of the batteries are defective in the sample of 20 should be more 0.99^20 = 0.8179. So, the probability that at least one defective is selected in a sample of 20 should be less than 1.0-0.8179 = 0.1821. This indicates that dispite all 20 out of twenty batteries in the sample are checked and found correct, still there is a probability that at least one defective may exist is 0.1821 for any lot of 20. Therefore, the test procedure should prescribe that if one defective found in the sample check should be sufficient to reexamine the whole process of manufacture and correct the process to improve to attain the desired efficient production.

Statistical quality control is a method which makes it possible to estimate the likely quality of a large lot of products or components on the basis of inspection of just a small sample. Mathematical techniques of statistical analysis are used to to determine the sampling plans that result in assuring that the conclusions drawn about the quality of the whole lot based on the quality of the sample. The sample plans are designed to achieve a specified permissible level of maximum defective percentage, and a level of confidence about the conclusion based on samples being correct. The sample plans specifies the size of the sample to be drawn, how the sample is to be drawn, the maximum level of defective that a sample may have without the whole lot being rejected. The sampling plan also specifies the action being taken in case quality based on sample is found to be unacceptable. These may include alternatives such as rejecting the whole lot, or inspecting the whole. The design of such sampling plans is best done by a qualified statistician. Actually, sampling plans to suit a variety of different requirements fave been worked out by experts and are available for use by others in form of standard tables of sampling plans.

I do not intend to provide a sampling plan for the situation described in question. However, I would like to point out that a sample of 20 is too small to make meaningful estimate about quality of lot for which maximum permissible defective probability is 0.01. With this sample size even if the the actual probability 2.5 times the maximum permissible, a sample of 20 will show no defective in about 50 percents of such samples drawn.