S4 June 2011 Q4

4. A proportion \(p\) of letters sent by a company are incorrectly addressed and if \(p\) is thought to be greater than 0.05 then action is taken.

Using \(\mathrm{H}_0 : p = 0.05\) and \(\mathrm{H}_1 : p \gt 0.05\), a manager from the company takes a random sample of 40 letters and rejects \(\mathrm{H}_0\) if the number of incorrectly addressed letters is more than 3.

Table 1 below gives some values, to 2 decimal places, of the power function of this test.

Table 1

A visiting consultant uses an alternative system to test the same hypotheses. A sample of 15 letters is taken. If these are all correctly addressed then \(\mathrm{H}_0\) is accepted. If 2 or more are found to have been incorrectly addressed then \(\mathrm{H}_0\) is rejected. If only one is found to be incorrectly addressed then a further random sample of 15 is taken and \(\mathrm{H}_0\) is rejected if 2 or more are found to have been incorrectly addressed in this second sample, otherwise \(\mathrm{H}_0\) is accepted.

Figure 1 shows the graph of the power function of the test used by the consultant.