NH, AN, T-Test and F-Test / 原假设，备择假设，T 检验和 F 检验

NH: the difference is 0.
AH: It is contradictory to NH.

检验统计量的目标是确定模型与数据的拟合程度。一个通俗的例子，你去服装店买衣服，衣装模特会告诉你衣服应该是什么样子， 这就是 the theoretical model 理论模型；你试穿同一件衣服，这就是 the data-based model 基于数据的模型。然后你需要确认你穿上衣服和模特穿上衣服的效果是否显著不同，这就需要求解 significant level 显著性水平，也就是进行 t 检验和 F 检验。

t 检验是比较两个不同组的均值的检验统计量。
F 检验是将我们计算的模型与平均值进行比较。如果它高于一个临界值，那么模型比平均值更善于解释数据。

引入几个概念：
sum of squares: a measure of how the residuals compare to the model or the mean, depending on which one we are working with.

The sum of squares of the residuals (SSR) : the sum of the squares of the residuals between the data points and the actual regression lines. They are squared to compensate for the negative values / 数据点和拟合模型之间的残差的平方和

The sum of squares of the total (SST) : the sum of the squares of the residuals between the data points and the mean of the sample / 数据点与样本之间的残差的平方和

The difference between the two (SSR – SST) will tell you the overall sum of squares (SSM) for the model itself. This is what we are after in order to finally start to calculate the actual F value.

The next step in the F-test process is to calculate the mean of squares for the residuals and for the model.

MSM: mean of squares of the model

MSR: mean of squares of the residuals

This calculation gives you a ratio of the model’s prediction to the regular mean of the data. Then you compare this ratio to an F-distribution table as you would the t-statistic. If the calculated value exceeds the critical value in the table, then the model is significantly different from the mean of the data, and therefore better at explaining the patterns in the data.

Ref
T-Test and F-Test: Fundamentals of Test Statistics