Define, in terms of \(\mathrm { H } _ { 0 }\) and/or \(\mathrm { H } _ { 1 }\),
the size of a hypothesis test,
the power of a hypothesis test.
The probability of getting a head when a coin is tossed is denoted by \(p\).
This coin is tossed 12 times in order to test the hypotheses \(\mathrm { H } _ { 0 } : p = 0.5\) against \(\mathrm { H } _ { 1 } : p \neq 0.5\), using a 5\% level of significance.
Find the largest critical region for this test, such that the probability in each tail is less than 2.5\%.
Given that \(p = 0.4\)
find the probability of a type II error when using this test,
find the power of this test.
Suggest two ways in which the power of the test can be increased.