the size of a test.
Jane claims that she can read Alan's mind. To test this claim Alan randomly chooses a card with one of 4 symbols on it. He then concentrates on the symbol. Jane then attempts to read Alan's mind by stating what symbol she thinks is on the card. The experiment is carried out 8 times and the number of times \(X\) that Jane is correct is recorded.
The probability of Jane stating the correct symbol is denoted by \(p\).
To test the hypothesis \(\mathrm { H } _ { 0 } : p = 0.25\) against \(\mathrm { H } _ { 1 } : p > 0.25\), a critical region of \(X > 6\) is used.
Find the size of this test.
Show that the power function of this test is \(8 p ^ { 7 } - 7 p ^ { 8 }\).
Given that \(p = 0.3\), calculate
the power of this test,
the probability of a Type II error.
Suggest two ways in which you might reduce the probability of a Type II error.