Question

There are two coins – one unbiased with
P(H) = 1/2, the other biased with
P(H) = 2/5
One of these coins is selected and tossed 5 times. If the head comes up at least twice,
the coin is assumed to be unbiased. Find the level of significance and power of the
test.

Answer 1

Null hypothesis is that coin is unbiased. Alternative hypothesis is that coin is biased.

Level of significance is probability to reject true null hypothesis:

$$\alpha=P_0(less\ than\ two\ heads)=P_0(0 \ heads)+P_0(1\ heads)=$$ $$=\dbinom{5}{0}({1 \over 2})^0(1-{1 \over 2})^{5-0}+\dbinom{5}{1}({1 \over 2})^1(1-{1 \over 2})^{5-1}=$$ $${3 \over 16}=0.1875$$

Power of the test is the probability to reject null hypothesis while alternative is true:

$$\alpha=P(less\ than\ two\ heads)=P(0 \ heads)+P(1\ heads)=$$ $$=\dbinom{5}{0}({2 \over 5})^0(1-{2 \over 5})^{5-0}+\dbinom{5}{1}({2 \over 5})^1(1-{2 \over 5})^{5-1}=$$ $$={1053 \over 3125}=0.33696$$