統計學（二）- 唐麗英（管院）
===
`NCTU` `IEM`
# Syllabus
- 這門課是Applied Statistics！！想要知道一堆公式推導的大神們，請選擇應數系「統計學(下學期開課)」、「高等統計學(上學期開課)」。
- 參考書目
- 上課用書
Introduction to Probability and Statistics 14/e (Mendenhall, Beaver, Beaver)
- 建議閱讀 (樓上那本有時候講得太簡略，以下按推荐順序排列)
- Mathematical Statistics and Data Analysis 3/e (John A. Rice)
(Mathematical Statistics第一順位用書)
- Introduction to Mathematical Statistics 7/e (Hogg, McKean, Craig)
(Mathematical Statistics第二順位用書)
- Fundamentals of Probability with Stochastic Processes 3/e (Saeed Ghahramani)
(資工系Probability用書)
- 老師超級親切，課程品質非常高，可以學會怎麼應用在生活中，加簽必須第一天就要去。
- 上課使用老師自製講義，搞清楚內容考試就90分無難度。
- 期末學期成績可得極高分，這一門3學分的課。
- 總共3次大型考試，第二次期中考是三個老師共同出題，學期中有若干次小考。
- 每個章節都要交手寫習題作業。可以做小組project當作加分用，題目自訂。

# Course
## Chapter 9~10 Test of Hypothesis：Part A - Concept
test of hypothesis(假說檢定)主要目的
1. 採集樣本資訊
2. 使用統計工具產生數值指標
3. 下結論，告訴我們到底要不要相信假說

基本上，test of hypothesis就是一種使用統計的SOP

:::danger
Five Steps for Test of Hypothesis (整學期會反覆應用，請務必宇宙級精熟)
1. Set up **Null Hypothesis** $H_0$ and **Alternative Hypothesis** $H_a$
2. Choose the **Level of Significance** $\alpha$
3. Calculate the **Test Statistics**
4. Choose the **Rejection Region**
5. Make a conclusion that is either **Reject $H_0$** or **Don't Reject $H_0$**
:::

接下來，我們來逐一詳細了解以上五步驟在幹嘛
`1.` Set up **Null Hypothesis** $H_0$ and **Alternative Hypothesis** $H_a$
Null Hypothesis $H_0$：the hypothesis against $H_a$
Alternative Hypothesis $H_a$：the hypothesis that we hope to support and become the truth

:::success
**答題小技巧**
1. $H_a$只會出現$>$、$\neq$、$<$這三種關係符號
2. 把題幹敘述的疑問句變成肯定句就是$H_a$
3. 題幹出現「claim that」字樣，$H_a$就是反對那個claim
4. 研究者做這麼多工作，就只是為了證明$H_a$是真的！
:::

`2.`Choose the **Level of Significance** $\alpha$
Type I Error：Reject $H_0$ when it's actually true.
Type II Error：Do not reject $H_0$ when it's actually false.

The probability of committing **Type I Error** is $\alpha$.
The probability of committing **Type II Error** is $\beta$.

通常我們會把$\alpha$設立為5%，也可以是1%、10%或是其他任意數值。

`3.`Calculate the **Test Statistics**
開始按計算機，算一堆mean、standard error、$z$、$t$、${\chi}^2$、$f$之類的值。
視乎做何種檢定而有不同的值要計算，公式忘了要回去翻以前的章節。

`4.`Choose the **Rejection Region**
Rejection Region (R.R.)白話的說就是region of rejecting $H_0$。
因為test of hypothesis會套用一種distribution model，所以會把$H_a$的boundary轉成那一個distribution model之下的值，稱作**Critical Value**。

**One-tailed Test**
- Left-tailed Test：$H_a$內含$<$
- Right-tailed Test：$H_a$內含$>$
例如$H_a: \mu > 1000$畫成圖就是這樣：
斜線部分表示如果$H_a$成真，代表samples出現在那個區域

**Two-tailed Test**
$H_a$內含$\neq$，例如$H_a: \mu \neq 1000$畫成圖就是這樣：
斜線部分表示如果$H_a$成真，代表samples出現在那個區域

`5.`Make a conclusion that is either **Reject $H_0$** or **Don't Reject $H_0$**
看看test statistics有無落在R.R.裡面，這樣子就知道要不要reject $H_0$了；也可以利用**p-value**和**power of the test**來綜合判斷要不要降低或者調高 $\alpha$

[**p-value**](https://en.wikipedia.org/wiki/P-value)
[**power of the test**](https://en.wikipedia.org/wiki/Statistical_power)

EXAMPLE：
:::info
Philips的廣告宣稱：所有貼有公司保固貼紙的燈泡，都具有5000小時以上的使用壽命。CFCT的研究員想要了解到底有沒有廣告不實，所以隨機到各個大賣場購買20個Philips的燈泡作為樣本，並且連續使用直到無法發出亮光。
測試結果如表格A，請問：請使用假說檢定法，判定Philips是否屬實？
1. Hypothesis：
$H_0: \mu_{life \ of \ bulbs} \geq 5000$
$H_a: \mu_{life \ of \ bulbs} < 5000$
2. $\alpha = 5 \%$
3. Test statistics：
$\bar{X} = 4995.8$、$s = 24.2$
Apply Student's t-test.
$\therefore t = \frac{\bar{X} - \mu_0}{s/\sqrt{n}} \approx -0.776$
4. Right-tailed Test：
$\nu = 19$
$critical \ value = -1.729$
$p$-$value \approx 22.4 \%$
5. Conclusion：
Do not reject $H_0$, so we don't have sufficient evidence to say that the claim in the advertisement is not truth.
:::
上面的例子有看不懂的部分，接下來的章節會一一補齊，只需要熟悉步驟順序以及內容就好。

## Chapter 9~10 Test of Hypothesis：Part B - Test Models
### Case 1：SINGLE Populations
以下**Five Steps**全部一樣，只有test statistics需要使用不同的model

- Test of $\mu$, when we have known $\sigma^2$
Apply [**standard normal distribution**](https://en.wikipedia.org/wiki/Normal_distribution#Standard_normal_distribution).
$Z = \frac{\bar{X} - \mu_0}{\sigma/\sqrt{n}}$ , where $\mu_0$ is the null-hypothesized value of mean

- Test of $\mu$, when we have **NOT** known $\sigma^2$
Apply [**student's t distribution**](https://en.wikipedia.org/wiki/Student%27s_t-distribution).
$t = \frac{\bar{X} - \mu_0}{s/ \sqrt{n}}$ , where $\nu = n-1$ and $n$ is the sample size

- Test of $P$
Apply [**standard normal distribution**](https://en.wikipedia.org/wiki/Normal_distribution#Standard_normal_distribution).
$Z = \frac{\hat{p} - P_0}{\sqrt{\frac{P_0(1-P_0)}{n}}}$
where $P_0$ is the null-hypothesized value of "success" proportion in Bernoulli trial

- Test of $\sigma^2$
Apply [**chi-squared distribution**](https://en.wikipedia.org/wiki/Chi-squared_distribution).
$\chi^2 = \frac{(N-1)s^2}{\sigma^2_0}$, where $\sigma^2_0$ is the null-hypothesized value of variance and $\nu = n-1$

### Case 2：TWO INDEPENDENT Populations
以下**Five Steps**全部一樣，只有hypothesis要小調整和test statistics需要使用不同的model

- Test of $\mu_1- \mu_2$, when we have known $\sigma^2_1$ and $\sigma^2_2$ (Cat.A)
$H_0: \mu_1- \mu_2$ ==$\geq \ = \ \leq$== $\delta_0$
$H_a: \mu_1- \mu_2$ ==$< \ \neq \ >$== $\delta_0$
where $\delta_0$ is the null-hypothesized difference of two [parameters](https://en.wikipedia.org/wiki/Parameter#Statistics_and_econometrics)
Apply [**standard normal distribution**](https://en.wikipedia.org/wiki/Normal_distribution#Standard_normal_distribution).

- Test of $\mu_1- \mu_2$, when we have NOT known $\sigma^2_1$ and $\sigma^2_2$
- Case of assumption **$\sigma^2_1 = \sigma^2_2$** (Cat.B)
$H_0: \mu_1- \mu_2$ ==$\geq \ = \ \leq$== $\delta_0$
$H_a: \mu_1- \mu_2$ ==$< \ \neq \ >$== $\delta_0$
Apply [**student's t distribution**](https://en.wikipedia.org/wiki/Student%27s_t-distribution).
where  and $\nu = (n_1-1)+(n_2-1)$
- Case of assumption **$\sigma^2_1 \neq \sigma^2_2$** (Cat.C)
$H_0: \mu_1- \mu_2$ ==$\geq \ = \ \leq$== $\delta_0$
$H_a: \mu_1- \mu_2$ ==$< \ \neq \ >$== $\delta_0$
Apply [**student's t distribution**](https://en.wikipedia.org/wiki/Student%27s_t-distribution).
where $\nu = \frac{(A+B)^2}{\frac{A^2}{(n_1-1)}+\frac{B^2}{(n_2-1)}}$ and $A=\frac{s^2_1}{n_1}$ is $SE(\bar{X_1})$ and $B=\frac{s^2_2}{n_2}$ is $SE(\bar{X_2})$

- Test of $P_1-P_2$
- Case of zero null hypothesis
$H_0: P_1- P_2$ ==$\geq \ = \ \leq$== $0$
$H_a: P_1- P_2$ ==$< \ \neq \ >$== $0$
Apply [**standard normal distribution**](https://en.wikipedia.org/wiki/Normal_distribution#Standard_normal_distribution).
where  and $X_1$ is the number of "success" events in Bernoulli trial
- Case of nonzero null hypothesis
$H_0: P_1- P_2$ ==$\geq \ = \ \leq$== $\delta_0$
$H_a: P_1- P_2$ ==$< \ \neq \ >$== $\delta_0$
Apply [**standard normal distribution**](https://en.wikipedia.org/wiki/Normal_distribution#Standard_normal_distribution).

- Test of $\frac{\sigma^2_1}{\sigma^2_2}$
$H_0: \sigma_1$ ==$\geq \ = \ \leq$== $\sigma_2$
$H_a: \sigma_1$ ==$< \ \neq \ >$== $\sigma_2$
Apply [**Fisher–Snedecor distribution**](https://en.wikipedia.org/wiki/F-distribution).
$F = \frac{s^2_1}{s^2_2}$, where $(\nu_1, \ \nu_2) = (n_1-1, \ n_2-1)$

### Case 3：TWO DEPENDENT Populations
以下**Five Steps**全部一樣，只有test statistics需要使用不同的model

### Flow Chart for Test of $\mu_1-\mu_2$

## Chapter 11 The Analysis of Variance (ANOVA)
### Part A：CRD ANOVA

### Part B：RDB ANOVA

## Chapter 12 Linear Regression and Correlation Analysis

# Homework

# Exam