t 检验 (t-test)

当总体 已知， 未知时，我们用样本方差来估计标准误，用估计标准误作为 估计值。
When the overall is known and is unknown, we estimate the standard error using the sample variance and use the estimated standard error as the estimate.

t 统计量为：
t-statistic is:

即：

适用规则：
Applicable rules:

1. 当 值已知, 用 分数。
When the value of is known, use to score the value of .

2. 当 值未知, 用 来估计 ，用 统计量。
When the value of is unknown, estimate with , and use the statistic.

1. 自由度(degree of freedom)：描述了样本中可以自由变化的分数的数目。

• Degrees of freedom describe the number of scores in the sample that are free to vary.

2. 若样本容量为n，因为样本均值对于样本中的分数值构成了限制，所以t检验中样本有df=n-1个自由度。

• If the sample size is n, the sample has df = n-1 degrees of freedom in the t-test because the sample mean poses a restriction on the value of the fraction in the sample.

3. t分布的形状受自由度df影响。n的数目越大(或df越大),t分布就越接近正态分布。

• The shape of the t-distribution is influenced by the degrees of freedom df. The larger the number of n (or the larger the df), the closer the t-distribution is to a normal distribution.

1. t 分布表描述了几个不同的 t 分布。对于每一个不同自由度，都存在一个不同的 t 分布(即使当 df 变大时,差别实际上变得很小)。

• The t-distribution table describes several different t-distributions. For each of the different degrees of freedom, there is a different t-distribution (even though the difference becomes very small as df gets larger).

2. 表中的每一行都对应于不同的 t 分布，因表中没有足够的空间列出对应每个可能的 t 分数概率，t 分布表中列出的只是最常用的临界区域的 t 分数(即，对应于那些最常用的 α 水平) 。

• Each row of the table corresponds to a different t-distribution, because there is not enough space in the table to list the probabilities corresponding to each possible t-score. The t-distribution table lists only the t-scores for the most commonly used critical regions (corresponding to those most commonly used α levels).

1. t 检验属于一种推论统计方法，我们根据抽取样本来推测其代表的总体分布。
   • The t-test belongs to a method of inferential statistics in which we infer the overall distribution it represents based on the samples drawn.
2. t 检验的步骤 (Steps for t-test):
   1. 陈述H₀和H₁；确定显著性标准a；
      • State the H₀ and H₁ and determine the significance criteria.
   2. 确定检验是单尾还是双尾；
      • Determine if the test is one or two-tailed;
   3. 确定检验的自由度df；
      • Determining the degrees of freedom of the test;
   4. 程序计算或查表得到临界 t 分数；
      • Calculate with the program, or look up the table to get the critical t-score.
   5. 计算样本的实际t分数；
      • Calculate the actual t-score of the sample;
   6. 比较样本的实际 t 分数与临界 t 分数；
      • Compare the actual t-score of the sample with the critical t-score;
   7. 对H₀作出结论。
      • Conclude whether to accept H₀.
3. t_obs=计算出的 t 分数(the actual t-score)。
4. t_crit=表中的临界 t 分数(the critical t-score)。