For a hypothesis test with a probability of a Type II error of 60% and a probability of Type I error of 5%,which of the following statements is most accurate?

A:the power of the test is 40%,and there is a 5% probability that the test statistic will exceed the critical values.

B:there is a 95% probability that the statistic will be between the critical values if this is a two-tail test.

C:the power of the test is 55%,and the confidence level is 95%

D:there is a 5% probability that the null hypothesis will be rejected when actually true ,and the probability of rejecing the null when it is false is 40%

请问一下A和B为什么错了，书后的答案没有看懂，谢谢了

A. The power of the test is 40% 正确， 后半部分错误， Type I error 的定义是说rejection of the null hypothesis when it is actually true, 注意这里的precondition是说null hypothesis是true，也就是说在这个hypothesis是true的情况下， 有5%的概率test static 超过critical values。但是如果null hypothesis本身就是false的，那么 test static超过critical values就是Correct Decision，也就是Power of Test，也就是说在这种情况下， 超出critical values的概率是40%。

所以到底超过critical values的概率是多少在不知道Null hypothesis 到底是true还是false的情况下是无法知道的。

B的错误的原因同A。