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.