=====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.