5.5 A firm receives components from a supplier, which it uses in its own production. The components are delivered in batches of 2000. The supplier claims that there are only 1% defective components on average from its production. However, production occasionally becomes out of control and a batch is produced with 10% defective components. The firm wishes to intercept these low-quality batches, so a sample of size 50 is taken from each batch and tested. If two or more defectives are found in the sample then the batch is rejected.

(a) Describe the two types of error the firm might make in assessing batches of components.

(b) Calculate the probability of each type of error given the data above.

(c) If instead, samples of size 30 were taken and the batch rejected if one or more rejects were found, how would the error probabilities be altered?

(d) The firm can alter the two error probabilities by choice of sample size and rejection criteria. How should it set the relative sizes of the error probabilities (i) if the product might affect consumer safety? (ii) if there are many competitive suppliers of components? (iii) if the costs of replacement under guarantee are high?