Answer to Question #86943 in Statistics and Probability for Anand

Question #86943
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.
1
Expert's answer
2019-03-25T12:06:28-0400

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"


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