Answer in Statistics and Probability for Anand #86832
A decision rule is a procedure that the researcher uses to decide whether to accept or reject the null hypothesis.
Two types of errors can result from a decision rule.
Type I error. A Type I error occurs when the researcher rejects a null hypothesis when it is true. The probability of committing a Type I error is called the significance level, and is often denoted by α.
Type II error. A Type II error occurs when the researcher accepts a null hypothesis that is false. The probability of committing a Type II error is called Beta, and is often denoted by β. The probability of not committing a Type II error is called the Power of the test.
We have Binomial distribution for every coin
“P(X=x)=dbinom{n}{x}p^x(1-p)^{n-x}”
Null hypothesis is that coin is unbiased (head comes up at least twice). Alternative hypothesis is that coin is biased.
Level of significance is probability to reject true null hypothesis:
“alpha=P_I(less than two heads)=P_I(0 heads)+P_I(1 heads)”
“alpha=dbinom{5}{0}({1over 2})^0(1-{1 over 2})^{5-0}+dbinom{5}{1}({1over 2})^1(1-{1 over 2})^{5-1}”
“alpha={1 over 32}+{5 over 32}={3 over 16}=0.1875”
Power of the test is the probability to reject null hypothesis while alternative is true:
“1-beta=P(less than two heads)=P(0 heads)+P(1 heads)”
“1-beta=dbinom{5}{0}({2over 5})^0(1-{2 over 5})^{5-0}+dbinom{5}{1}({2over 5})^1(1-{2 over 5})^{5-1}”
“1-beta={243 over 3125}+{810 over 3125}={1053 over 3125}=0.33696”
