Statistics 简明教程
Statistics - Power Calculator
每当进行一个假设检验时，我们需要确定该检验是高质量的。检验一个检验的功效或灵敏度的一种方法是计算当备择假设正确时，检验能正确地否定原假设的概率。换句话说，检验的功效是在它是正确时接受备择假设的概率，其中备择假设检测统计检验中的一个效应。
Whenever a hypothesis test is conducted, we need to ascertain that test is of high qualitity. One way to check the power or sensitivity of a test is to compute the probability of test that it can reject the null hypothesis correctly when an alternate hypothesis is correct. In other words, power of a test is the probability of accepting the alternate hypothesis when it is true, where alternative hypothesis detects an effect in the statistical test.
检验的功效也可以通过检验第一类错误 (${ \alpha }$)和第二类错误 (${ \beta }$)的概率来检验，其中第一类错误表示错误地否定了有效的原假设，而第二类错误表示错误地保留了无效的原假设。第一类或第二类错误的可能性越小，则统计检验的功效越大。
Power of a test is also test by checking the probability of Type I error($ { \alpha } $) and of Type II error($ { \beta } $) where Type I error represents the incorrect rejection of a valid null hypothesis whereas Type II error represents the incorrect retention of an invalid null hypothesis. Lesser the chances of Type I or Type II error, more is the power of statistical test.
Example
对学生进行了一项调查以检验他们的智商水平。假设对 16 名学生的随机样本进行测试。检验者检验原假设（学生的智商为 100）与备择假设（学生的智商不为 100），使用显著性水平为 0.05 和标准差为 16。如果真实总体均值为 116，则假设检验的功效是多少？
A survey has been conducted on students to check their IQ level. Suppose a random sample of 16 students is tested. The surveyor tests the null hypothesis that the IQ of student is 100 against the alternative hypothesis that the IQ of student is not 100, using a 0.05 level of significance and standard deviation of 16. What is the power of the hypothesis test if the true population mean were 116?
Solution:
Solution:
由于原假设下的检验统计分布遵循学生 t 分布。此处 n 很大，我们可以用正态分布来近似 t 分布。由于犯第一类错误的概率 (${ \alpha }$)为 0.05，当检验统计量 ${ T \ge 1.645 }$ 时，我们可以否定原假设 ${H_0}$。让我们使用以下公式计算使用检验统计量的样本均值。
As distribution of the test statistic under the null hypothesis follows a Student t-distribution. Here n is large, we can approximate the t-distribution by a normal distribution. As probability of committing Type I error($ { \alpha } $) is 0.05 , we can reject the null hypothesis ${H_0}$ when the test statistic $ { T \ge 1.645 } $. Let’s compute the value of sample mean using test statistics by following formula.
让我们使用以下公式计算统计检验的功效。
Let’s compute the power of statistical test by following formula.
因此，我们有 99.09% 的机会否定原假设 ${H_0: \mu = 100 }$，有利于备择假设 ${H_1: \mu \gt 100 }$，其中未知总体均值为 ${ \mu = 116 }。
So we have a 99.09% chance of rejecting the null hypothesis ${H_0: \mu = 100 } $ in favor of the alternative hypothesis $ {H_1: \mu \gt 100 } $ where unknown population mean is $ {\mu = 116 } $.