本章內容：
1. 介紹Experimental Design 和 ANOVA（Analysis of Variance）
2. 搞清楚：這一章的學習，學了什麼？在現實世界有什麼應用價值？
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
問題：爲什麼 SST = SSE + SSTR ?
方差分析的基本原理是認爲不同處理組的均數間的差別基本來源有兩個：
(1) 實驗條件，即不同的處理造成的差異，稱爲組間差異。用變量在各組的均值與總均值之偏差平方和的總和表示，記作SSb，組間自由度dfb。
(2) 隨機誤差，如測量誤差造成的差異或個體間的差異，稱爲組內差異，用變量在各組的均值與該組內變量值之偏差平方和的總和表示， 記作SSw，組內自由度dfw。
總偏差平方和 SSt = SSb + SSw。
用上面的例子：
1 . Between-Treatments Estimate就是 SSTR 由於三種方法帶來效率的不同，導致在method A B C之間得到的測量值的差異
mean square due to treatments
2. Within-Treatments Estimate 就是 SSE即由於隨機誤差的原因使得各組內部的測量值各不相等 mean square due to error
3. Treatment 感覺就是不同方法
13.3 Multiple Comparison Procedures
Fisher’s LSD
之前的f-test得出，三個population mean 不相等，但是跟隨着的問題是：是1 3不相等？還是1 2不相等？還是2 3 不相等？
可以用之前學過的t test來做
如果用LSD方法來做的話，非常快，效果非常好，只需要簡單的比較兩個sample mean的差別和LSD，誰大誰小
問題： LSD怎麼推導出來的？相當於用之前的 t-test，換了一下格式
問題：這裏的 t-test和 cha10的 t-test 公式有點不一樣？爲什麼可以用 MSE？
Type I Error Rates
Comparisonwise Type I error Rate : indicate the level of significance associated with a single pairwise comparison. 在一個test裏面的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
我們要解決的問題是：這三種方法對於 controller stress的影響是不是有不同？
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
當我們需要得到有關兩個或兩個以上因子，同時發生時的統計結論時，Factorial Experiment 是值得考慮的。
It is an experimental design that allows simultaneous conclusions about two or more factors.
案例：對於 GMAT 考試，學校準備了3種複習方案，考試會有三個學院的學生參與。因此有九種情況。
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.