Multi-stage or conditional testing

3 The manufacturer of electronic components uses the following process to test the proportion of defective items produced. A random sample of 20 is taken from a large batch of components.

• If no defective item is found, the batch is accepted.
• If two or more defective items are found, the batch is rejected.
• If one defective item is found, a second random sample of 20 is taken. If two or more defective items are found in this second sample, the batch is rejected, otherwise the batch is accepted.

The proportion of defective items in the batch is denoted by \(p\), and \(q = 1 - p\).

1. Show that the probability that a batch is accepted is \(q ^ { 20 } + 20 p q ^ { 38 } ( q + 20 p )\). For a particular component, \(p = 0.01\).
2. Given that a batch is accepted, find the probability that it is accepted as a result of the first sample.

4. A poultry farm produces eggs which are sold in boxes of 6 . The farmer believes that the proportion, \(p\), of eggs that are cracked when they are packed in the boxes is approximately 5\%. She decides to test the hypotheses $$\mathrm { H } _ { 0 } : p = 0.05 \text { against } \mathrm { H } _ { 1 } : p > 0.05$$ To test these hypotheses she randomly selects a box of eggs and rejects \(\mathrm { H } _ { 0 }\) if the box contains 2 or more eggs that are cracked. If the box contains 1 egg that is cracked, she randomly selects a second box of eggs and rejects \(\mathrm { H } _ { 0 }\) if it contains at least 1 egg that is cracked. If the first or the second box contains no cracked eggs, \(\mathrm { H } _ { 0 }\) is immediately accepted and no further boxes are sampled.

1. Show that the power function of this test is $$1 - ( 1 - p ) ^ { 6 } - 6 p ( 1 - p ) ^ { 11 }$$
2. Calculate the size of this test. Given that \(p = 0.1\)
3. find the expected number of eggs inspected each time this test is carried out, giving your answer correct to 3 significant figures,
4. calculate the probability of a Type II error. Given that \(p = 0.1\) is an unacceptably high value for the farmer,
5. use your answer from part (d) to comment on the farmer's test.

A bank has detection software that can be set at two different levels, 'Mild' and 'Severe'. • When it is set at Mild, 0.1% of all transactions are queried. • When it is set at Severe 0.5% of all transactions are queried.

1. One day the bank has 500 000 transactions. The software is set on 'Mild'. There are 480 false positives. Only \(\frac{1}{4}\) of the unauthorised transactions are queried. Complete the table. [3]
2. What is the ratio of false positives to false negatives? [1]
3. If the software had been set on 'Severe' for the same set of 500 000 transactions, with the total numbers of authorised and unauthorised transactions the same as in part (i) of this question, the number of false negatives would have been 5. What would the ratio of false positives to false negatives have been with this setting? [3]