1. A manufacturer produces boxes of screws containing short screws and long screws. The manufacturer claims that the probability, \(p\), of a randomly selected screw being long, is 0.5

A shopkeeper does not believe the manufacturer's claim. He designs two tests, \(A\) and \(B\), to test the hypotheses \(\mathrm { H } _ { 0 } : p = 0.5\) and \(\mathrm { H } _ { 1 } : p < 0.5\) In test \(A\), a random sample of 10 screws is taken from a box of screws and \(\mathrm { H } _ { 0 }\) is rejected if there are fewer than 3 long screws. In test \(B\), a random sample of 5 screws is taken from a box of screws and \(\mathrm { H } _ { 0 }\) is rejected if there are no long screws, otherwise a second random sample of 5 screws is taken from a box of screws. If there are no long screws in this second sample \(\mathrm { H } _ { 0 }\) is rejected, otherwise it is accepted.

1. Find the size of test \(A\).
2. Find the size of test \(B\).
3. Find an expression for the power function of test \(B\) in terms of \(p\). Some values, to 2 decimal places, of the power function for test \(A\) and the power function for test \(B\) are given in the table below.

   \(p\)	0.1	0.2	0.3	0.4
   Power test \(A\)	0.93	\(r\)	0.38	0.17
   Power test \(B\)	0.83	0.55	0.31	0.15

4. Find the value of \(r\). The shopkeeper believes that the value of \(p\) is less than 0.4
5. Suggest which of the tests the shopkeeper should use. Give a reason for your answer.