====== t 检验 (t-test) ======

===== t 统计量 (t-statistic) =====

当总体 {{:第六章:6.6:mu.svg}} 已知，{{:第六章:6.6:sigma.svg}} 未知时，我们用样本方差来估计标准误，用估计标准误作为 {{:第六章:6.6:sigma.svg}} 估计值。\\
When the overall {{:第六章:6.6:mu.svg}} is known and {{:第六章:6.6:sigma.svg}} is unknown, we estimate the standard error using the sample variance and use the estimated standard error as the {{:第六章:6.6:sigma.svg}} estimate.

t 统计量为：\\
t-statistic is:

{{ :第六章:6.6:t1.svg |}}

即：

{{ :第六章:6.6:t2.svg |}}

===== t 与 z 的不同适用条件 (Different application conditions for t and z) =====

{{ :第六章:6.6:tandz.svg |}}

适用规则：\\
Applicable rules:

1. 当 {{:第六章:6.6:sigma2.svg}} 值已知, 用 {{:第六章:6.6:z.svg}} 分数。\\
When the value of {{:第六章:6.6:sigma2.svg}} is known, use {{:第六章:6.6:z.svg}} to score the value of {{:第六章:6.6:z.svg}}.

2. 当 {{:第六章:6.6:sigma2.svg}} 值未知, 用 {{:第六章:6.6:s2.svg}} 来估计 {{:第六章:6.6:sigma2.svg}}，用 {{:第六章:6.6:t.svg}} 统计量。\\
When the value of {{:第六章:6.6:sigma2.svg}} is unknown, estimate {{:第六章:6.6:s2.svg}} with {{:第六章:6.6:sigma2.svg}}, and use the {{:第六章:6.6:t.svg}} statistic.

===== t 统计量的自由度 (Degree of Freedom) =====

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.

===== t 分布表 (t-distribution table) =====

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).

{{::t分布表.png?400|}}

===== t 检验 (t-test) =====
  - 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.
  - t 检验的步骤 (Steps for t-test):
    - 陈述H<sub>0</sub>和H<sub>1</sub>；确定显著性标准a；
      * State the H<sub>0</sub> and H<sub>1</sub> and determine the significance criteria.
    - 确定检验是单尾还是双尾；
      * Determine if the test is one or two-tailed;
    - 确定检验的自由度**df**；
      * Determining the degrees of freedom of the test;
    - 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.
    - 计算样本的实际t分数；
      * Calculate the actual t-score of the sample;
    - 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;
    - 对H<sub>0</sub>作出结论。
      * Conclude whether to accept H<sub>0</sub>.
  - t<sub>obs</sub>=计算出的 t 分数(the actual t-score)。
  - t<sub>crit</sub>=表中的临界 t 分数(the critical t-score)。