=====Z检验的前提(The Premise of Z-test)===== **1.随机样本** * 原因:样本必须对总体有代表性,而随机取样有助于确保取样的代表性 。 **2.独立观察** * 原因:与样本代表性有关, 每个观察应该与所有其它观察是独立的。一个特定的观察的概率应当保持恒定。 **3.σ保持恒定** * 原因:在Z检验的处理中,对总体中的每一个个体都加上(或减去) 一个常数,总体的均值可能因处理而导致变化,但标准差不变。 **4.取样样本是相对正态的** * 原因:如果取样样本不能近似为正态分布,就无法将它们标准化得到对应的Z值。 * 相对正态的两种情况:①原总体分布就是正态分布;②由于中心极限定律可以近似为正态分布。 * 相对正态的例子:二项分布 **违反任一前提的后果:** 严重地危及依据样本对总体作出推论的有效性。 **1.Random sample** * Reason: The sample must be representative of the population, and random sampling helps ensure the representativeness of the sample **2.Independent observation** * Reason: It is related to the representativeness of the sample, and each observation should be independent of all other observations. The probability of a specific observation should remain constant. **3.σ maintain constant** * Reason: In the processing of Z-test, a constant is added (or subtracted) for each individual in the population, and the mean of the population may change due to processing, but the standard deviation remains unchanged. **4.The sample is sampled relatively normal** *Reason: If the sampled samples cannot be approximated as a normal distribution, they cannot be standardized to obtain the corresponding Z value. *There are two cases of relative normality: ① the original population distribution is a normal distribution; ② Due to the fact that the central limit law can be approximated as a normal distribution. *An example of relative normality: binomial distribution **Consequences of violating any premise:** Seriously endangers the validity of inferring the population based on the sample.