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