本章内容：
1. 介绍Experimental Design 和 ANOVA（Analysis of Variance）
2. 搞清楚：这一章的学习，学了什么？在现实世界有什么应用价值？
13.1 An Introduction of Experimental Design and Analysis of Variance
μ1=mean number of units produced per week using methodA
μ2=mean number of units produced per week using methodB
μ3=mean number of units produced per week using methodC
H0: μ1 = μ2 = μ3
Ha: Not all population means are equal
Assumptions for Analysis of Variance
- For each population, the response variable is normally distributed.
- The variance of the response variable, is the same for all the populations.
- The observations must be independent.
13.2 Analysis of Variance and the Completely Randomized Design
H0: μ1 = μ2 = μ3 =… = μk
Ha: Not all population means are equal
μj = Mean of the jth population
xij = value of observation i for treatment j
nj = number of observations for treatment j
xj = sample mean for treatment j
s2j = sample variance for treatment j
sj = sample standard deviation for treatment j
13.2.1 Between-Treatments Estimate of Population Variance
Mean Square due to Treatments(MSTR) =
If H0 is true, MSTR provides an unbiased estimate of σ2. However, if the means of the k populations are not equal, MSTR is not an unbiased estimate of σ2, MSTR should overestimate σ2.
13.2.2 Within-Treatments Estimate of Population Variance
Mean Square due to Error (MSE) = 
13.2.3 Comparing the Variance Estimates: The F Test
13.2.4 ANOVA Table
问题：为什么 SST = SSE + SSTR ?
方差分析的基本原理是认为不同处理组的均数间的差别基本来源有两个：
(1) 实验条件，即不同的处理造成的差异，称为组间差异。用变量在各组的均值与总均值之偏差平方和的总和表示，记作SSb，组间自由度dfb。
(2) 随机误差，如测量误差造成的差异或个体间的差异，称为组内差异，用变量在各组的均值与该组内变量值之偏差平方和的总和表示， 记作SSw，组内自由度dfw。
总偏差平方和 SSt = SSb + SSw。
用上面的例子：
1 . Between-Treatments Estimate就是 SSTR 由于三种方法带来效率的不同，导致在method A B C之间得到的测量值的差异
mean square due to treatments
2. Within-Treatments Estimate 就是 SSE即由于随机误差的原因使得各组内部的测量值各不相等 mean square due to error
3. Treatment 感觉就是不同方法
13.3 Multiple Comparison Procedures
Fisher’s LSD
之前的f-test得出，三个population mean 不相等，但是跟随着的问题是：是1 3不相等？还是1 2不相等？还是2 3 不相等？
可以用之前学过的t test来做
如果用LSD方法来做的话，非常快，效果非常好，只需要简单的比较两个sample mean的差别和LSD，谁大谁小
问题： LSD怎么推导出来的？相当于用之前的 t-test，换了一下格式
问题：这里的 t-test和 cha10的 t-test 公式有点不一样？为什么可以用 MSE？
Type I Error Rates
Comparisonwise Type I error Rate : indicate the level of significance associated with a single pairwise comparison. 在一个test里面的type I error，例如α = .05
Experimentwise Type I error rate: 1- (0.95)* (0.95)*(0.95) = 0.1426
13.4 Randomized Block Design
13.4.1 Air Traffic Controller Stress Test
我们要解决的问题是：这三种方法对于 controller stress的影响是不是有不同？
和之前不同的是，controllers are believed to differ substantially in their ability to handle stressful situations. What is high stress to one controller might be only moderate or even low stress to another.
所以之前的 MSE 包括，random error and error due to individual controller differences.
So we need to separate SST into three parts:
SST = SSTR + SSBL + SSE
13.5 Factorial Experiment
当我们需要得到有关两个或两个以上因子，同时发生时的统计结论时，Factorial Experiment 是值得考虑的。
It is an experimental design that allows simultaneous conclusions about two or more factors.
案例：对于 GMAT 考试，学校准备了3种复习方案，考试会有三个学院的学生参与。因此有九种情况。
需要研究的是：
1. Main effect (Factor A): Do the preparation programs differ in terms of effect on GMAT scores?
2. Main effect (factor B): Do the undergraduate colleges differ in terms of effect on GMAT scores?
3. Interaction effect (factor A and B): Do students in some colleges do better on one type of preparation program whereas others do better on a different type of preparation program?
SST = SSA + SSB + SSAB + SSE
a= number of levels of factor A
b = number of levels of factor B
r = number of replications
nT = total number of observations taken in the experiment, nT = abr
Therefore, the study provides no reason to believe that the three preparation programs differ in their ability to prepare students from the different col- leges for the GMAT.