**1.独立样本t检验的假设**
  * 虚无假设（H0：μ1-μ2=0）：独立样本所来自的两个总体的均值之间没有显著的差异，即所抽取的两个样本来自同一个总体。

  * 备择假设（H1：μ1-μ2≠0）：独立样本所来自的两个总体的均值之间有显著差异。

**2.总体方差的估计**
总体方差的合并估计值为{{ :spooled.png?120 |}}
  * 估计的思路：将两样本进行__加权平均__，权重是样本的__自由度__
  * 估计的前提：两样本方差大体相等（即满足__方差同质性__）

**3.均值分布的方差的计算**
  * {{ :s1.png?80 |}}    
  * {{ :s2.png?80 |}}
  * 计算的思路：由于样本容量n存在差异，两个样本均值的分布不一定相同，因此均值分布的变异性需要考虑__样本容量__。

**4.标准误的计算**
  * {{ :s1-22.png?150 |}}
  * {{ :s1-2.png?150 |}}
  * 计算的思路：均值差异样本的方差是总体1与总体2的均值分布的方差之和。

**5.t统计量的计算**
  * {{ :t独立计算.png?300 |}}

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**与单样本T检验有三点不同之处**
  - 比较的分布是均值差异的分布（X1-X2）
  - 确定t的临界值是基于两个样本的自由度（df1+df2）
  - 比较分布的样本分数是基于两个分数之差


**1.Hypothesis of independent sample t-test**
  *The null hypothesis (H0: μ1- μ2=0): There is no significant difference between the means of the two populations from which the independent samples come, indicating that the two samples are from the same population.
  *Alternative hypothesis (H1: μ1- μ2≠0): There is a significant difference between the means of two populations from which independent samples come.

**2.Estimation of population variance**
The combined estimated value of the overall variance is {{ :spooled.png?120 |}}
  *Estimation approach: Perform a __weighted average__ of two samples, with the weight being the __ degrees of freedom of the samples__.
  *Assumption for estimation: The variance of the two samples is roughly equal (i.e. __satisfies the homogeneity of variance__)

**3.Calculation of variance of mean distribution**
  * {{ :s1.png?80 |}}    
  * {{ :s2.png?80 |}}
  *Calculation approach: Due to differences in sample size n, the distribution of the mean values between the two samples may not be the same. Therefore, the variability of the mean distribution needs to consider the __sample size__.

**4.Calculation of standard error**
  * {{ :s1-22.png?150 |}}
  * {{ :s1-2.png?150 |}}
  *Calculation idea: The variance of the mean difference sample is the sum of the variances of the mean distributions of population 1 and population 2.

**5.Calculation of t-statistic**
  * {{ :t独立计算.png?300 |}}

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**There are three differences from single sample T-test**

  - The distribution of comparison is the distribution of mean difference (X1-X2)
  - The determination of the critical value of t is based on the degrees of freedom of two samples (df1+df2)
  - The sample scores for comparing distributions are based on the difference between two scores